Course Title: Statistics for Food Scientist
Course code: FST 301
By
R. A. Atuna
ratuna@uds.edu.gh
1
Course outline
• Basic definitions and fundamental concepts
–
–
–
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Statistics
Why stats in research
validity and reliability
Sampling and sampling techniques
• Data, Measures and Collection Methods
• Basic experimental design
– Parametric Data analysis
– Fitting statistical models to the data
•
•
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Test of association
Mean comparison
Single factor analyses
Factorial analyses
– Non Parametric Data Analysis
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• Objectives
The objectives of the FST 301 course include, but not limited to the
following:
– To train students to independently design experiments
– To collect and enter data and analyse appropriately
– To train students on handling parametric data
– To train students on handling non-parametric data from sensory analysis
• Learning outcomes
– On successful completion of this course, students should be able to:
– Design and conduct research
– Collect data, clean, analyse, present and interpreted results
• Mode of delivery
– Lecture, tutorials, and seminars
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• Research is the“the systematic investigation into existing or new knowledge, to
establish or co firm facts, reaffirm the results of previous work, solve new or
existing problems, support theorems, or develop new theories”.
– Leads to the advancement of knowledge
• Systematic investigation (Burns, 1997) or inquiry whereby data are collected,
analyzed and interpreted in some way in an effort to "understand, describe,
predict or control an educational or psychological phenomenon or to empower
individuals in such contexts" (Mertens, 2005, p.2).
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Statistics
• It is the study of the collection, organization,
analysis, and interpretation of data.
• It also deals with the planning of data
collection, in terms of the design of surveys
and experiments.
5
Statistics
• It is the study of the collection, organization, analysis, and interpretation
of data.
• It also deals with the planning of data collection, in terms of the design of
surveys and experiments.
Why Statistics in Research
• Statistics assists researches to be understood, to describe phenomena and
to help draw reliable conclusions about these phenomena.
• In every research there will be some results, sometimes numerical or
logical statement.
• Statistics checks the accuracy, consistency and acceptability of the resul
ts obtained in a research based on probability measures.
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• It provides a platform for research as to; how to go about the
research, the techniques to use in data collection, how to go about
the data description, analysis and he inference to be made.
• It plays the role of an arbiter in research it has power and influence
to make judgements and decide on what to be done for the
acceptability of research.
• Statistics does not decide any research conclusion(s) but assists in
reaching the conclusion(s).
• It stand as a bedrock (principles) on which any research is based.
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Factors to be considered in choosing a research method
• Research questions
– To answer my research question, should I or do I need to . . .
•
•
•
•
•
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Make causal inferences?
Generalize from the cases studied (sample) to a broader group (population)?
Study change over time?
Interact with subjects/participants?
Generate my own data sources and data?
Use more than one design?
• Ethics
– Thinking about how the research impacts on those involved with the research
process.
– Ethical research should gain informed consent, ensure confidentiality, be legal and
ensure that respondents and those related to them are not subjected to harm.
– All this needs to be weighed up with the benefits of the research.
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• Budget
• Time
– As a general rule, the more in-depth the method the more time
consuming it is. Also, doing your own primary research tends to take
longer than using secondary sources.
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Validity and Reliability
Validity
Appropriateness of the methods to the research Question
• Addresses whether your research explains or measures what you said
you would be measuring or explaining
• Validity of interpretations of the data it must be a product of analysis–
Internal validity
• Representativeness of sample – Content validity
• Generalizability to a population – External validity
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Reliability
• Internal consistency – Cronbach alpha
• How closely related a set of items are as a group
• Test Retest
• Whether similar results are obtained when the same participants
respond to the same test a second time.
• Consistency of measure.
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Research experimental design
• The plan a researcher will follow in conducting a study.
• The art of planning and executing experiments.
• The greatest strength of an experimental research design, is largely due
to random assignment, is its internal validity.
– the degree to which the results are attributable to the independent variable
and not some other rival explanation.
• The greatest weakness of experimental designs may be external validity.
– The extent to which the results of a study can be generalized
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Design of experiment begins with
– Determining the objectives of an experiment
– Selecting the process factors for the study.
An experimental design is the laying out of a detailed experimental
plan in advance of doing the experiment.
Experiment
– A series of test or a test in which purposeful changes one made to input
variables of a process or system so we may observe and identify reasons for
changes that may be observed in the output response.
– A study undertaken in which the researcher has control over some
conditions in which the study takes place and control over (some aspects
of) the independent variables being studied.
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Survey
• A research design in which a sample of subjects is drawn from a
population and studied (usually interviewed) to make inferences
about a population.
Census
• A survey of an entire population in contrast to a survey of a sample
drawn from a population.
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Natural experiment
• A situation happening naturally, that is, without the researcher’s
manipulation that approximates an experiment.
• Variables occur naturally in such a way that they have some of the
characteristics of the control and experimental groups.
– E.g a solar eclipse provides astronomers opportunities to observe the sun that they
cannot provide for themselves by experimental manipulation.
– the comparison has been made between the number of dental cavities in a city
with naturally occurring fluoridated water and a similar city without fluoridation.
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Quasi-experiment:
• A type of research design for conducting studies in field or real life
situations where the researcher may be able to manipulate some
independent variables but cannot randomly assign subjects to control
and experimental groups.
• For example, you cannot cut off someone’s unemployment benefits
to see how well he or she could get along without them or to see
whether an alternative job-training program would be more effective
for some unemployed persons. But you could try to find volunteers
for the new program.
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Descriptive research:
• Research that describes phenomena as they exist. Descriptive research is
usually contrasted with experimental research in which experiments are
controlled and subjects are given different treatments.
Control group
• In experimental research, a group that, for the sake of comparison, does
not receive the treatment the experimenter is interested in.
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Experimental group
• A group receiving some treatment in an experiment.
• Data collected about people in the experimental group are
compared with data about people in a control (who received no
treatment) and/or another experimental group (who received a
different treatment).
Experimental unit
• The smallest independently treated unit of a study.
– E.g. – if 90 subjects were randomly assigned to 3 treatment groups, the
study would have three experimental units, and not 90.
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Treatment
• In experiments, a treatment is what researchers do to subjects in
the experimental group but not to those in the control group. A
treatment is thus an independent variable.
Factor
• In analysis of variance, an independent variable, that is, a variable
presumed to cause or influence another variable.
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Random
• Said of events that are unpredictable because their occurrence is
unrelated to their characteristics.
• The chief importance of randomness in research is that, by using it,
researchers increase the probability that their conclusions will be
valid.
• Random assignment increases internal validity.
• Random sampling increases external validity.
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Random assignment
• Putting subjects into experimental and control groups in such a way
that each individual in each group is entirely assigned by chance.
• In other words, each subject has an equal probability of being
placed in each group. Using random assignment reduces the
likelihood of bias.
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Sampling Methods
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Uses of probability sampling
• There are multiple uses of probability sampling:
– Reduce Sample Bias: Using the probability sampling method, the bias in
the sample derived from a population is negligible to non-existent. The
selection of the sample mainly depicts the understanding and the
inference of the researcher. Probability sampling leads to higher
quality data collection as the sample appropriately represents the
population.
– Diverse Population: When the population is vast and diverse, it is essential
to have adequate representation so that the data is not skewed towards
one demographic. For example, if Square would like to understand the
people that could make their point-of-sale devices, a survey conducted
from a sample of people across the US from different industries and socioeconomic backgrounds helps.
– Create an Accurate Sample: Probability sampling helps the researchers
plan and create an accurate sample. This helps to obtain well-defined data.
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• Nonrandom Sampling Techniques
– Convenience sampling (i.e., it simply involves using the people who are the most
available or the most easily selected to be in your research study).
– Quota sampling (i.e., setting quotas and then using convenience sampling to obt
ain those quotas).
– Purposive sampling (i.e., the researcher specifies the characteristics of the popul
ation of interest and then locates individuals who match those characteristics).
– Snowball sampling (i.e., each research participant is asked to identify other pote
ntial research participants who have a certain characteristic).
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Methods of data collection
• Tests (personality, aptitude, achievement, and performance)
• Questionnaires (i.e., self report instruments).
• Interviews (i.e., situations where the researcher interviews the participants).
• Focus groups (i.e., a small group discussion with a group moderator present to ke
ep the discussion focused).
• Observation (i.e., looking at what people actually do).
• Existing or Secondary data (i.e., using data that are originally collected and then a
rchived or any other kind of “data” that was simply left behind at an earlier time f
or some other purpose).
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Guidelines for Designing Experiments
To use the statistical approach in designing and analysing an
experiment, it is necessary for everyone involved in the experiment to
have a clear idea in advance of exactly what is to be studied, how the
data are to be collected, and at least a qualitative understanding of
how these data are to be analysed.
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Recognition of and statement of the problem
• This may seem to be a rather obvious point, but in practice often neither it is
simple to realize a problem requiring experimentation exist, nor is it simple to
develop a clear and generally accepted statement of this problem.
• It is necessary to develop ideas necessary to develop all ideas about the
objectives of the experiment.
• Usually, it is important to solicit input from all concerned parties: engineering,
quality assurance, manufacturing, marketing, management, customer, and
operating personnel (who usually have much insight and who are too often
ignored).
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Selection of the response variables
• In selecting the response variable, the experimenter should be certain that this
variable really provides useful information about the process under study.
• Most often, the average or standard deviation (or both) of the measured
characteristic will be the response variable.
• It is usually critically important to identify issues relayed to defining the
responses of interest and how they are to be measured before conducting the
experiment.
• Sometimes designed experiments are employed to study and improve the
performance of measurement systems.
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Choice of factors levels (treatments) and range
• When considering the factors that may influence the performance of a
process or system, the experimenter usually discovers that these factors
can be classified as either potential design factors or nuisance factors.
• The potential design factors are those factors that the experimenter may
wish to vary in the experiment.
• Often we find that there are a lot of potential design factors, and some
further classification of them is helpful.
– Designed factors: The design factors are the factors actually selected for study in the
experiment.
– held-constant factors: Held-constant factors are variables that may exert some
effects on the response, but for purposes of the present experiment these factors
are not of interest, so they will be held at a specific level.
– allowed-to –vary factors.
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Choice of experimental design
• If the above pre-experimental planning activities are done correctly, this step is
relatively easy.
• Choice of design involves consideration of sample size (number of replicates),
selection of a suitable run order for the experimental trials, and determinations
of whether or not blocking or other randomization restrictions are involved.
• In selecting a design, it is important to keep the experimental objectives in mind.
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Performing the experiment
• When running the experiment, it is vital to
monitor the process carefully to ensure that
everything is being done according to plan.
• Error in experimental procedure at this stage will
usually destroy experimental validity.
• Up-front planning is crucial to success. It is easy to
underestimate the logistical and planning aspects
of running a designing experiment in a complex
manufacturing or research and development
environment.
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Statistical analysis of the data
• Statistical methods should be used to analyse the data so that results
and conclusions are objective rather than judgemental in nature.
• If the experiment has been designed correctly and performed according
to the design, the statistical method required is not elaborate.
• Often we find that simple graphical methods play an important role in
data analysis and interpretation. Because many of the questions that the
experimenter wants to answer can be cast into a hypothesis-testing
framework, hypothesis testing and confidence interval estimation
procedures are very useful in analysing data from designed experiment.
• Remember that statistical methods cannot prove that a factor (factors)
has a particular effect. They only provide guidelines as the reliability and
validity of results.
• When properly applied, statistical methods do not allow anything to be
proved experimentally, but they do allow us to measure the likely error
in a conclusion or to attach a level of confidence to a statement.
• The primary advantage of statistical method is that they add objectivity
to the decision-making process. Statistical techniques usually lead to
sound conclusions.
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Conclusion and recommendations
• Once the data have been analysed, the
experimenter must draw practical conclusions
about the results and recommend a course of
action.
• Graphical methods are often useful in this stage,
particularly in presenting the results to others.
• Follow-up runs and confirmation testing should
also be performed to validate the conclusions from
the experiment.
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Lecture 2
Data, Measures and Collection
Methods
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Variables and Measurements
• Statistics is essentially concerned with the application of logic
and objectivity in the understanding of events.
• Necessarily, the events have to be identified in terms of relevant
characteristics or occurrences that could be measured in
numeric or non-numeric expressions.
• The characteristic or occurrence is technically referred to as a
variable because it may assume different values or forms within
a given range of values or forms known as the domain of the
variable.
• The act of attaching values or forms to variables is known as
measurement.
• A variable may be qualitative or quantitative; and a quantitative
variable may be continuous or discontinuous (i.e. discrete).
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Quantitative Variable
• A quantitative variable Y is numeric. It may assume
values y that can be quantified. Age, weight, height,
volume are all quantitative variables. Production,
prices, costs, sales, sizes, etc., are also quantitative.
• A quantitative variable Y may be coded qualitatively
• E.g – In an agronomic trial, maize plot response may
be coded A for yields exceeding 7500 kg/ha; B for
yields in the bracket 6001 – 7500 kg/ha; and C for
yields between 5001 and 6000 and D for yields 5000
and below.
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Qualitative Variable
• A qualitative variable X may have forms x that could only be
described but could not be measured numerically.
• It is a non-numeric entity.
– Nutrient when identified only as nitrogen, phosphorus, potassium or
zinc; thus, “Nutrient” is a qualitative variable or factor. Other
examples are “variety” of maize; “sex” of a farmer; “country” of
origin, etc.
– Yield scores when recorded as high, moderate or low is qualitative.
• A qualitative variable has states or forms, which can be
described, categorised, classified, or qualified.
• A qualitative X may be recorded as if a quantitative variable
– E.g – in a farm survey, a male farmer may be coded as 0 and a female
as 1.
– In an entomological experiment, response may be classified as 0 if the
plant survives a chemical treatment and 1 if otherwise.
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Continuous and Discrete Variables
• A quantitative variable may be continuous or
discontinuous i.e. discrete.
• The family size of farming household, X, in a crop
survey may take between values 1, 2, 3, 4, 5, 6, 7, etc.
• In the interval (0, 100) the number of values which X
can take is countable (finite), it is 101; but the size of
farms Y cultivated by the household may take an
uncountable (infinite) number of values within even
the shortest of intervals when measurement is not
limited by degree of accuracy nor by the precision of
the measuring equipment. Family size variable X is a
typical discrete variable while farm size Y is a typical
continuous one.
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Scale of Measurement
• Any form of measurement falls into four
categories (or scales):
– Nominal,
– Ordinal,
– Interval, and
– Ratio
The scale of measurement will ultimately dictate
the statistical procedures (if any) that can be used
in processing the data.
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Nominal Scale of Measurement
• The word nominal comes from the Latin nomen, meaning “name”. Hence,
we can “measure” data to some degree by assigning names to them. For
example, we can measure a group of children by dividing groups: boys &
girls; overweight, normal weight & underweight.
• Nominal measurement is quite simplistic, but it does divide data into
discrete categories that can be compared with one another.
• We have alluded to situations where a basically qualitative variable is
measured (coded) in numeric terms. Socio-economic classes namely
lower, middle and upper may be coded as 1, 2, 3, male and female
gender as 1 and 2; rural, semi- urban and urban locations as 1, 2 and 3
respectively.
• Only a few statistical procedures are appropriate for analysing nominal
data. We can use the mode as an indicator of the most frequently
occurring category within our data set; for instance, we might determine
that there are more males than females in the second Year BMB class in
UDS.
41
Ordinal Scale of Measurement
• With an ordinal scale of measurement, can think in terms of the symbols
“greater than” (>) and “less than” (<). We can compare various pieces of
data in terms of one being greater or higher than another. In essence, this
scale allows us to rank-order our data (hence its name ordinal).
• We can roughly measure level of education on an ordinal scale by
classifying people as being unschooled or having an elementary, high
school, college, or graduate education.
• The ordinal level is the second weakest level of measurement; it does not
permit use of the four arithmetic operations.
• An ordinal scale expands the range of statistical techniques we can apply to
the data.
• We can also determine the median or halfway point in a set of data.
• We can use a percentile rank to identify the relative position of any item or
individual in a group.
• We can determine the extent of relationship between two characteristics
by the means of Spearman’s rank order correlation.
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Interval Scale of Measurement
• An interval scale of measurement is characterised by two
features:
– It has equal units of measurement and
– Its zero point has been established arbitrarily.
• E.g – The Fahrenheit (F) and Celsius (C) scales.
• The intervals between any two successive numbers of degrees
reflect equal changes in temperature but the zero point is not
equivalent to a total absence of heat.
• Because an interval scale reflects equal distance among adjacent
points any statistics that are calculated using addition or
subtraction (for instance means, standard deviations and
Pearson’s product moment correlation) can now be used.
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Ratio Scale
• Two measurement instruments may help you understand
the difference between interval and ratio scales: a
thermometer or a yard stick.
• If we have a thermometer that measures temperature on
either the Fahrenheit or Celsius scale, we CANNOT say that
80°F is as twice as warm as 40°F. Why? Because these
scales do not originate from a point of absolute zero; a
substance may have some degree of heat even though its
measured temperature falls below zero.
• With a yard stick, however, the beginning of a linear
measurement is absolutely the beginning. If we measure a
desk from the left edge to the right edge, that’s it.
• A measurement of “zero” means there is no desk there at
all, and a “minus” distance is not even possible.
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• More generally, a ratio scale has two
characteristics:
– Equal measurement units (similar to an interval scale)
and
– An absolute zero point, such that zero on the scale
reflects a total absence of the quantity being
measured.
– Only ratio scales allow us to make comparisons that
involve multiplication or division.
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We can summarise our description of the four scales
this way: if you can say that
• One object is different from another you have a
nominal scale;
• One object is bigger or better or more of anything
than another, you have an ordinal scale;
• One object is so many units (degrees or inches)
more than another, you have an interval scale;
• One object is so many times as big or bright or tall
or heavy as another, you have a ratio scale.
46
Interval scales
Non-interval scales
Scale
Characteristics of the Scale
Statistical Possibilities of the Scale
Table 1- A summaryMeasurement
of measurement
scales, their
characteristics and their statistical
implications
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Nominal scale
A scale that “measures” in terms of names or
designations of discrete units or categories
Enables one to determine the mode, the
percentage values, or the chi-square
Ordinal scale
A scale that “measures” in terms of such values
as “more” or “less,” “larger or smaller,” but
without specifying the size of the intervals
Enables one also to determine the median,
percentile rank, and rank correlation
Interval scale
A scale that measures in terms of equal
intervals or degrees of difference, but whose
zero point, or point of beginning, is arbitrarily
established
Enables one also to determine the mean,
standard deviation, and product moment
correlation; allows one to conduct most
inferential statistical analyses
Ratio scale
A scale that measures in terms of equal
intervals and an absolute zero point of origin
Enables one also to determine the geometric
mean and the percentage variation; allows one
to conduct virtually any inferential statistical
analysis
Lecture 3
Basic experiment design
48
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Data analysis
• When data are qualitative, analysis involves
looking at your data graphically to see what the
general trends are, and also fitting statistical
models to it. Some methods for analysis include
the following:
– Frequency distribution (Histogram)
Once you have collected some data a very useful thing to do
is to plot a graph of how many times each score occurs.
This is known as a frequency distribution, or histogram,
which is a graph plotting of values of observations on the
horizontal axis, with a bar show many times each value
occurred in the data set.
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Frequency distributions come in many different
shapes and sizes. It is quite important, therefore, to
have some general descriptions for common types of
distributions.
In an ideal world our data would be distributed
symmetrically around the centre of all scores. This is
known as a normal distribution
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• There are two main ways in which a distribution
can deviate from normal:
– lack of symmetry-called skew and
– Pointyness-called kurtosis
A positively (left figure) and negatively (right figure) skewed distribution
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Distributions with positive kurtosis (leptokurtic, left figure) and negative kurtosis (platykurtic,
right figure)
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The centre of a distribution
• We can also calculate where the centre of a
frequency distribution lies, known as the central
tendency. There are three measures commonly
used: the mean, the mode and the median.
– The Mean
• The mean is the measure of central tendency that you are most
likely to have heard of because it is simply the average score and
the media are full of average scores. To calculate the mean, we
simply add up all the scores and then divide by the total number
of scores we have. We can write is in equation form as: 𝑀𝑒𝑎𝑛 =
𝑛
𝑖=1 𝑋𝑖
; where the numerator simply means, “add up all the
scores”; Xi just means “ a particular score, and the n
(denominator) the total number of scores.
𝑛
54
The Mode
• The mode is simply the score that occurs most frequently in the
data set. This is easy to spot in a frequency distribution because it
will be the tallest bar.
• To calculate the mode, simply arrange the data points in ascending
(or descending) order; count how many times each score occurs,
and the scores that occurs most is the mode. One problem with
mode is that it can often take on several values.
The Median
• Another way to quantify the centre of a distribution is to look for
the middle score when scores are ranked in order of magnitude.
This is called the median.
• To calculate the median, we first arrange these scores into
ascending order.
• Next, we find the position of the middle score by counting the
number of scores we have collected (n), adding 1 to this value, and
then dividing by 2. that is (n + 1)/2 .
• Then we find the score that is positioned at the location we have
just calculated.
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The dispersion in a distribution
• It can also be interesting to try to quantify the
spread, or dispersion, of scores in the data. The
easiest way to look at dispersion is to subtract the
smallest score from the largest score. This is known
as the range of scores.
• One problem with the range is that because it uses
only the highest and the lowest scores, it is affected
dramatically by extreme scores.
– E.g. using the following data, 108, 103, 252, 121, 93, 57,
40, 53, 22, 116 and 98. The range is 252-22 = 230.
– By dropping the largest score, the range will be 99, which
is less than half the size.
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• One way around this problem is to calculate the
range when we exclude values at the extremes of
the distribution.
• One convention is to cut off the top and bottom
25% of scores and calculate the range of the
middle 50% of scores – known as the interquartile
range.
57
A summary of measures of Central Tendency and Variability
Measure
How it’s determined
(N = number of scores; X = individual
score, and M = Mean)
Data for which it is appropriate
Measures of Central Tendency

The most frequently occurring score is
identified
Median

The scores are arranged in order from
smallest to the largest, and the middle

score when N is an odd number or the
midpoint between to middle scores when
N is an even number
Mean
All the scores are added together, and
their sum is divided by the total number
(N) of scores
Range
The difference between the highest and
lowest scores in the distribution
Measures of variability
Interquartile range
Standard deviation
The difference between the
percentiles
𝑆=
𝑋−𝑀
𝑁
25th and
Data on interval and ratio scales
Data that fall in a normal
distribution

Data o ordinal, interval, and ratio
scales

Data on ordinal, interval, and ratio
scales
Especially useful for highly skewed
data
Data on interval and ratio scales
Most appropriate for normally
distributed data
75th

2
𝑆2 =
𝑋−𝑀
𝑁
2
Data on ordinal, interval, and ratio
scale
Data that are highly skewed






Variance
58

Mode
Data on nominal, ordinal, interval,
and ratio scales
Multimodal distributions - two or
more modes may be identified
when a distinction has multiple
peaks

Data on interval and ratio scales
Most appropriate for normally
distributed data
Especially useful in inferential
statistical procedures (e.g.,
analysis of variance)
EXPLORING THE DATA
Exploratory Data Analysis (EDA)
• After the data are entered into spreadsheet and
imported to any statistical software (e.g.,
GENSTAT, SPSS, Minitab, SAS, etc) or entered
directly into the statistical software, the first step
to complete (before running any inferential
statistics) is EDA, which involves computing
various descriptive statistics and graphs.
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• Exploratory Data Analysis is used to examine and get
to know your data. EDA is important to do for
several reasons:
– To see if there are problems in the data such as outliers,
non-normal distributions, problems with coding, missing
values, and/or errors inputting the data.
– To examine the extent to which the assumptions of the
statistics that you plan to use are met.
– To get basic information regarding the demographics of
subjects to report in the method section or results
section.
– To examine relationships between variables to determine
how to conduct the hypothesis-testing analyses.
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• There are two general methods used for EDA:
generating plots of the data and generating
numbers from your data. Both are important and
can be very helpful methods of investigating the
data.
• Descriptive Statistics (including the minimum,
maximum, mean, standard deviation, and
skewness),
frequency
distribution
tables,
boxplots, histograms, and stem and leaf plots are
a few procedures used in EDA.
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Box and Whisker Plots
• In a box-and-whisker plot (or boxplot), there is a box
that spans the inter quartile range of the values in
the variable, with a line indicating the median.
Whiskers are drawn extending beyond the ends of
the box as far as the minimum and maximum
values.
• The whiskers are lines, and extend only to the inner
"fences" which are shown as continuous lines. The
upper line is defined as the upper quartile and the
lower line is defined as the lower fence.
• Thus a boxplot is the graphical representation of a
5-number summary of a dataset: minimum, Q1,
median, Q3, maximum.
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Box and Whisker Plots
63
Uses of boxplot
• Comparing groups
– Boxplots are a useful tool to compare groups of data. In
Fig. 3.11 for instance, it looked as if the yield of the new
variety is higher than the yield of the standard variety.
There is however some overlap and remember the scale
we are working with (minimum value is 1.7 tons/ha,
maximum value is 2.5). A formal statistical test has to
confirm the difference, but if this test shows completely
different results to the graph, we know something is
wrong.
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• Outliers
65
Shape of distribution
• A boxplot gives an idea about the shape of the
distribution, although you can also get this
information from other plots.
66
Stem & Leaf Diagram
• This draws a stem and leaf plot from a numeric
data. The plot comprises stems, indicating leading
digits, and leaves indicating subsequent digits.
• Stem-and-leaf plots are a method for showing the
frequency with which certain classes of values
occur.
• You could make a frequency distribution table or
a histogram for the values, or you can use a stemand-leaf plot and let the numbers themselves to
show pretty much the same information.
67
• For instance, suppose you have the following
list of values: 12, 13, 21, 27, 33, 34, 35, 37, 40,
40, 41. You could make a frequency
distribution table showing how many tens,
twenties, thirties, and forties you have:
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Frequency
Class
Frequency
10 - 19
2
20 - 29
2
30 - 39
4
40 - 49
3
• You could make a histogram, which is a bargraph showing the number of occurrences, with
the classes being numbers in the tens, twenties,
thirties, and forties:
69
• The downside of frequency distribution tables
and histograms is that, while the frequency of
each class is easy to see, the original data points
have been lost. You can tell, for instance, that
there must have been three listed values that
were in the forties, but there is no way to tell
from the table or from the histogram what those
values might have been.
70
• On the other hand, you could make a stemand-leaf plot for the same data:
The "stem" is the left-hand column which contains the tens digits. The "leaves" are the lists in
the right-hand column, showing all the ones digits for each of the tens, twenties, thirties, and
forties.
71
How to check for errors
• Look over the raw data (questionnaires, interviews, or observation
forms) to see if there are inconsistencies, double coding, obvious
errors, etc. Do this before entering the data into the computer.
• Check some, or preferably all, of the raw data (e.g., questionnaires)
against the data in your SPSS Data Editor file to be sure that errors
were not made in the data entry.
• Compare the minimum and maximum values for each variable in your
Descriptive output with the allowable range of values in your
codebook.
• Examine the means and standard deviations to see if they look
reasonable, given what you know about the variables.
• Examine the N column to see if any variables have a lot of missing
data, which can be a problem when you do statistics with two or more
variables. Missing data could also indicate that there was a problem in
data entry.
• Look for outliers in the data. Check the Assumptions
72
Statistical Assumptions
• Every statistical test has assumptions.
• They are much like the directions for appropriate use
of a product found in an owner's manual.
• Assumptions explain when it is and isn't reasonable to
perform a specific statistical test.
• When t-test was developed, for example, the person
who developed it needed to make certain assumptions
about the distribution of scores, etc., in order to be
able to calculate the statistic accurately.
• If these assumptions are not met, the value that
software calculates, which tells the researcher whether
or not the results are statistically significant, will not be
completely accurate and may even lead the researcher
to draw the wrong conclusion about the results.
73
Parametric tests:
• These include most of the familiar ones (e.g., analysis of
variance, Pearson correlation, etc). They usually have more
assumptions than nonparametric tests. Parametric tests were
designed for data that have certain characteristics, including
approximately normal distributions.
• Some parametric statistics have been found to be "robust" to
one or more of their assumptions. Robust means that the
assumption can be violated without damaging the validity of
the statistic.
– E.g, one assumption of ANOVA is that the dependent variable is
normally distributed for each group.
• Statisticians who have studied these statistics have found that
even when data are not completely normally distributed (e.g.,
they are somewhat skewed), they still can be used.
74
Nonparametric tests
• These tests (e.g., chi-square, Kruskal-Wallis,
Mann-Whitney U, Spearman rho) have fewer
assumptions and often can be used when the
assumptions of a parametric test are violated.
For example, they do not require normal
distributions of variables or homogeneity of
variances.
75
Lecture 4 – 7
Fitting statistical models to the data
76
Hypotheses
• There are two very important hypotheses that are used
when analysing data.
• These are scientific statements that can be split into
testable hypotheses.
• The hypothesis or prediction that comes from theory is
usually saying that our effect will be present. This
hypothesis is called the alternative hypothesis, and is
denoted by H1.
• The other type of hypothesis is known as the null
hypothesis (H0). This hypothesis is the opposite of the
alternative hypothesis and so would usually state that an
effect is absent.
– E.g.- Alternative hypothesis (H1): Days after planting of orangefleshed sweetpotato have an effect on β-carotene.
– Null hypothesis (H0): Days after planting will not affect the βcarotene of orange-fleshed sweetpotato.
77
• The reason that we need the null hypothesis is
because we cannot prove the experimental
hypothesis (alternative hypothesis) using statistics,
but we can reject the null hypothesis.
• If our data give us confidence to reject the null
hypothesis then this provides support for our
experimental hypothesis.
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Types of analyses
• There are numerous types of analyses you can
apply depending the data set you have. In this
course, we will consider the following:
• Test of associations
– Regression & Correlation
• Comparing two means
• Single factor analysis of variance
– One-way ANOVA- Completely Randomised Design
(CRD)
– Randomised Complete Block design
• Factorial design
79
Regression and correlation analyses
• In many problems, two or more variables are related,
and it is of interest to model and explore this
relation.
Types of association
• Association between response variables (outcome
variables). Example, acidity of maize and pH of the
dough
• Association between response and treatment. When
treatments are quantitative, such as different
fermentation times, and the acidity of maize dough.
It is possible to describe the association between the
treatment and response.
• Association between response and environment
80
• There are two tests of association
 Regression
 Correlation
Correlation
• It provides a measure of the degree of association between the variables or the
goodness of fit of a prescribed relationship to the data at hand.
– Correlation coefficient (r)
• The quantity r, called the linear correlation coefficient, measures the strength and the direction of a
linear relationship between two variables.
• The value of r is such that -1 < r < +1.
• The + and – signs are used for positive linear correlations and negative linear correlations,
•
•
•
•
81
respectively.
Positive correlation: If x and y have a strong positive linear correlation, r is close to +1. An r value of
exactly +1 indicates a perfect positive fit.
Negative correlation: If x and y have a strong negative linear correlation, r is close to -1. An r value of
exactly -1 indicates a perfect negative fit.
No correlation: If there is no linear correlation or a weak linear correlation, r is
close to 0. A value near zero means that there is a random, nonlinear relationship
between the two variables
A perfect correlation: of ± 1 occurs only when the data points all lie exactly on a
straight line. If r = +1, the slope of this line is positive. If r = -1, the slope of this
line is negative.
The value of r ranges between ( -1) and ( +1)
The value of r denotes the strength of the association
as illustrated by the following diagram.
strong
-1
intermediate
-0.75
-0.25
weak
weak
0
indirect
perfect
correlation
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intermediate
0.25
strong
0.75
1
Direct
no relation
perfect
correlation
83
• A correlation greater than 0.8 is generally described as strong,
whereas a correlation less than 0.5 is generally described
as weak.
Coefficient of Determination, r 2 or R2
• The coefficient of determination, r2, is useful because it gives the
proportion
of
the variance (fluctuation) of one variable that is predictable
from the other variable.
• It is a measure that allows us to determine how certain one can
be in making predictions from a certain model/graph.
• The coefficient of determination represents the percent of the
to the line of best fit.
– E.g. – if r = 0.922, then r 2 = 0.850, which means that
85% of the total variation in y can be explained by the linear
relationship between x and y (as described by the regression
equation).
84
A sample of 6 children was selected, data about
their age in years and weight in kilograms was
recorded as shown in the following table. It is
required to find the correlation between age and
weight.
serial
No
85
Age
(years)
Weight
(Kg)
1
2
3
7
6
8
12
8
12
4
5
5
6
10
11
6
9
13
Importance of Correlation
• Correlation helps us in determining the degree of relationship
between variables. It enables us to make our decision for the future
course of actions.
• Correlation analysis helps us in understanding the nature and degree
of relationship which can be used for future planning and forecasting.
• Forecasting without any prior correlation analysis may prove to be
defective, less reliable and more uncertain.
Limitation of Correlation
• r is a measure of linear relationship only
• Correlation does not always imply causality.
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Regression analysis
• It describes the effect of one or more variables
(designated as independent variables) on a single
variable (designated as the dependent variable) by
expressing the latter as a function of the former.
• For the analysis, it is important to clearly distinguish
between the dependent and independent variable, a
distinction that is always not obvious.
• The purpose of this analysis is to examine how
effectively one or more variables allow you to
predict the value of another (dependent) variable.
87
Simple linear regression
• A simple linear regression generates an equation in which a single
independent variable yields a prediction for the dependent.
• For simple linear regression, the following conditions must hold:
– There is only one independent variable X affecting the dependent variable Y.
– The relationship between Y and X is known, or can be assumed, to be linear.
Multiple linear regression
• A multiple linear regression yields an equation in which two or more
independent variables are used to predict the dependent variables.
– The relationship between two variables is linear if the change is constant
throughout the whole range under consideration
– It is a straight line and represented by the equation (y = mx + c)
• To predict the yield Y, for a given X, the value of X, should be in the range
of X min to X max. You cannot extrapolate the line outside this range.
88
• Assumptions of Multiple linear regression:
– A linear relationship between the dependent and independent variables.
– The independent variables are not highly correlated with each other.
– The variance of the residuals is constant.
– Independence of observation.
– Multivariate normality
89
Classifications
• Regression and correlation analyses can be
classified into four types:
Simple linear regression and correlation
Multiple linear regression and correlation
Simple non linear regression and correlation
Multiple non linear regression and correlation
Solve examples practically using GenStats
90
Comparing Two Means
• Often we have two unknown means and are
interested in comparing them to each other.
• Ho: no differences between the populations.
Paired t-test- for comparing paired or matched data.
E.g. - “before and after” sometimes also called
“longitudinal” studies also produce pair data. Each patient
contributes two paired observation: thus before values and
after values.
Two sample t-test
• Comparing two independent samples- this is also called
“two sample” t- test.
• Here treatment A or B is applied to different subjects.
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One sample t-test
• Comparing a single mean to a referenced
mean- some experiments involve comparing
only one population mean μ to a specified
value, say, μo. The hypotheses are
– Ho: μ = μo
– H1: μ ≠ μo
92
Single factor analysis
• Experiments in which a single factor varies whiles all
others are kept constant are called single-factor
experiments.
• Treatments consists solely of the different levels of the
single variable factor
• All other factors are applied uniformly to all plots at a
single prescribed level.
– Example, testing the effect of different fermentation times
(say, 1-day, 2-day and 3-day duration) on microbial
population in pito is a single-factor experiment.
– Only the fermentation duration differs from one
experiment to another, and all others kept constant (type
of millet, level of inoculums added, temperature the work
is held, etc)
93
• There are two groups of experimental design that
are applicable to a single-factor experiment
• One group is the family of complete block
designs, which is suited for experiments with a
small number of treatments and is characterized
by blocks, each of which contains at least one
complete set of treatments. Example, is the
Completely Randomised Design (CRD)
• The other group is the family of incomplete block
designs, suited for large number of treatments
and is characterized by blocks, each of which
contains only a fraction of the treatments to be
tested.
94
Analysis of variance for CRD
• Two sources of variations among the n observation
• One is the treatment; the other is experimental error
• A major advantage of the CRD is the simplicity in the
computation of its analysis of variance, especially when
the number of replications is not uniform for all the
treatments.
• The model for One-Way ANOVA:
Yij = µ + Ai + eij
Where:
Yij: jth observation in the ith treatment group
µ: a general mean (overall mean)
Ai: parameter unique to the ith treatment
eij: random residual error
95
Randomised Complete Blocks Design (RCBD)
• Here, there is one disturbance factor, namely the blocks.
• The experimental units are partitioned in blocks and each block is a
complete single replicate of the treatments.
• This is the connotation of the word ‘complete’; the randomisation of
treatments to units within blocks is done independently of other blocks
and treatment comparisons are made within blocks; hence variation
between blocks is eliminated.
• The Model for RCBD
The model is specified as Yij =  + Ai + Bj + eij
Where:
Yij: jth observation in the ith treatment group
µ: a general mean (overall mean)
Ai: parameter unique to the ith treatment
Bj: effect of the jth block
eij: random residual error
96
Factorial design
• Biological organisms are simultaneously exposed to many growth factors
during their lifetime.
• Because an organism’s response to any single factor may vary with level of
the other factors, single-factor experiments are often criticised for their
narrowness.
• A full factorial experiment is one where design consists of two or more
factors, each with discrete possible values or “levels”, and where
experimental units take on all possible combinations of these levels across
all such factors.
• Full factorial design may also be called a fully crossed design
• Such an experiment allows the investigator to study the effect of each
factor on the response variable, as well as the effects of interactions
between factors on the response variable
• For vast majority of factorial experiments, each factor has only two levels
• This is usually called 2x2 factorial design (22)
• It considers two levels (base) for each of two factors (the power or
superscript), or number of levels number of factors, producing 2x2=4 factorial
points
• When a design is denoted a 23 factorial, this identifies the number of
factors (3); and the levels for each factor is 2; and 8 (23 = 2 x 2 x 2)
experimental conditions are in the design (23 = 8)
97
• The model is specified as Yijk =  + Ai + Bj + Ai*Bj + eij
Yijk: kth observation in the ith treatment group (A) and jth treatment
group B
µ: a general mean (overall mean)
Ai: effect of the ith treatment group
Bj: effect of the jth treatment group
Ai*Bj: interaction between Ai and Bj
eij: random residual error
98