Data Management
Introduction
Data management is a process by which information is acquired and processed to ensure
the accessibility and reliability of the data for its users. One of the most important tool in
processing and managing such information is statistics. Statistics is utilized in most areas of
human endeavor. It is usually used in education, research, business, agriculture, and other fields
and even in everyday life activities.
Data or the pieces of information may be collected by conducting a survey, interview,
observation, and experiment. The data gathered can be properly organized and presented
graphically by a line graph, bar graph or pictograph or with the aid of a statistical table known as
frequency distribution table (FDT). A concise and meaningful conclusion is obtained from the
analysis and interpretation of data. Relevant information can be deduced from the analysis of
numerical descriptions and predictions may be made based on a small group to project the whole
population. The work of statistics offers a wide area of concern. Thus, statistics is subdivided into
two branches, namely: descriptive statistics and inferential statistics.
Statistics is a science which deals with the collection, organization, presentation, analysis, and
interpretation of data so as to give a more meaningful information.
Descriptive statistics refers to the collection, organization, summary, and presentation of data
while inferential statistics deals with the interpretation and analysis of data where conclusion is
drawn based from the subset of the population.
In descriptive statistics, a set of data is simply described without drawing any inferences
or implications. The data is merely summarized and discussed in a clear, concise and informative
manner. In inferential statistics, information or inferences concerning a large group known as
population is provided based on the study of a representative group or selected members in the
population which are identified as sample. Calculating the average rating of a class of 40 students
in Math 01illustrates the descriptive statistics while determining the performance of the same
class based on the performance of 10 randomly selected members in the class exhibits inferential
statistics.
BASIC TERMS
Some of the basic terminologies and notations involved in statistics are the following:
a. Population - a collection or set of things or objects under consideration
b. Sample - a subset or representative group of the population
c. Data - refers to the information gathered in a research Statistical data are classified
according to their sources, namely: primary data or secondary data.
 Primary data – information gathered from respondents by the researcher himself.
 Secondary data – information obtained from published materials or data gathered by
other individuals or agencies. These are the data which are transcribed from original
sources.
d. Array – listing of observations which are arranged in an increasing or decreasing
magnitude
e. Parameter - a value which is computed from a population
f. Statistic – a value which is computed from a sample
g. Variable – a characteristic of interest that has been observed or measured on every
member of the population or sample. A variable may be quantitative or qualitative where
quantitative variable is further classified as discrete or continuous.
i.
ii.
Quantitative/Numerical variable – describes the amount or number of an
element of a sample or population
 Discrete – takes on a countable amount (it is usually expressed as whole
number) Example: number of books owned by a student
 Continuous – measured in a continuous scale (it takes any value within a
range or interval) Example: height of the students (in feet)
ii. Qualitative/Categorical variable – describes the quality, category, or character
of an element of a population or sample Examples: gender (male or female) hair
color (black, brown, blonde) level of satisfaction of a student on his grade (highly
satisfied, satisfied, not satisfied)
Levels of Measurement
A more detailed distinction, termed as the levels of measurement, is used by some researchers
in examining the information that is collected. It is classified as follows:
1. Nominal Measurement - numbers or symbols are used to code or classify each element
in the population. Note that the assigned numbers have no numerical meaning. Examples:
gender, educational background, employment status
2. Ordinal Measurement– uses numerical category that expresses the meaningful order.
There is no indication of distance between positions. The numbers become meaningful
because they reveal whether one class or category is more or less than the other.
Categories are ranked according to the order of their value on the property like first,
second, third; oldest, next oldest, youngest. Example: rank in beauty contest
3. Interval Measurement– has equal intervals. There is significance to the distance between
any two values. It tells us that one unit differs by a certain amount of the property from
another unit. It has no absolute zero. Example: Aptitude test, temperature
4. Ratio Measurement – A variable measured at this level not only includes the concepts of
order and interval, but also includes the idea of ’nothingness’, or absolute zero. Example:
Measurement of height, weight, ages
Remark: The scale of measurement depends mainly on the method of measurements and not on
the property being measured. For instance, the weight of a pack of milk measured in kilograms
has an interval scale but if the boxes are labelled as one of small, medium or large, the weight is
measured in ordinal scale
Measure of Central Tendency
One way of summarizing the data is to figure out the data set by using the descriptive measures.
Among the most commonly used descriptive measures which are important are the measures of
central tendency and measures of dispersion.
A measure of central tendency (or central location) is a single value that is used to identify the
“center” of the data set or set of observations.
The three measures of central tendency are the mean, median and mode
The mean also known as the arithmetic average is the sum of all the observed values divided by
the number of observations in the data set. It can be computed as 𝜇 = 𝑋𝑖 𝑛 𝑖=1 𝑛 where 𝑥𝑖 is the
𝑖 𝑡ℎ observation and 𝑛 is the number of observations in the data set.
The mean of the population is symbolized by the lowercase letter “mu” in Greek alphabet,µ,
while the mean of the sample is represented by x̄ (x – bar).
Example 1:
The scores of five students who are selected randomly in a class of Math 01 are as follows: 44,
37, 41, 35 and 32. Find their average score.
Solution:
Applying the mean of ungrouped data gives
.
Hence, the average score of the five students is 37.8.
The means of subgroups can be combined to come up with the group mean known as
weighted mean. This can be calculated using the formula
Example 2: If the final examination of a class in statistics is given the weight 2, the average quizzes
the weight 3, and a project report the weight 1, what would be the mean grade of a student who
got the grades 90, 85 and 87, respectively.
Solution:
The mean grade of the student is 87.
Remarks:
1. The mean may not be an actual observation in the data set.
2. The mean reflects the magnitude of every observation since every observation
contributes to the value of the mean.
3. The mean is not a good measure of central tendency if there is an extreme value or
observation since it is easily affected by extreme values. The best measure of center for
this case is the median.
The median is a single value which divides an array of observations into two equal parts such that
50% of the observations falls above it and the remaining 50% falls below it. It may be written
symbolically by 𝑥̃ read as “x - tilde”.
The median of the data set consisting of an odd – numbered observations is the middlemost
value in the list. That is,
where n is the number of observations. If is even, the median is
the average of the two middlemost values. It can be computed as
where m1 and
m2 are the two middlemost values. Take note that the observations are first arranged in an array
form (from lowest to highest) before getting the median value. Example 1: The number of books
owned by the eleven children are as follows: 5, 2, 4, 6, 5, 10, 7, 6, 9, 8, 6. What is the median?
Solution:
Arrange the data in an array form: 2, 4, 5, 5, 6, 6, 6, 7, 8, 9, 10. Since the list contains 11 numbers
then the median is the middlemost value (6th number) which is 6.
Example 2: Compute the median of the data set: 2.5, 4.0, 5.8, 3.5, 2.5, 8.2, 7.1, 3.7
Solution:
Forming an array, we have 2.5, 2.5, 3.5, 3.7, 4.0, 5.8, 7.1, 8.2. There are values, hence, the median
is calculated as
.
Remarks:
1. The median value may not be an actual observation in the data set.
2. The median is a positional value, hence, it is not affected by the presence of extreme
observations.
3. When the data is qualitative, median is not a possible measure so described the center
by determining the mode
The mode is an observation that occurs most frequently in the given data set.
Example 1:
Find the mode in the following sets of scores.
a) set A: 36, 36, 12, 29, 35, 45. 50, 45, 45, 53
b) set B: 8, 7, 6, 5, 6, 9, 2, 3, 11, 11, 43, 10
c) set C: 39, 23, 25, 25, 63, 37, 45, 37, 48, 51, 28, 45, 50
d) set D: 2, 9, 8, 12, 5, 13, 6, 10
Solution:
The mode in set A is 45 because 45 occurs most frequently in the list. Both 6 and 11 have the
most number in set B, therefore, set B has the mode equal to 6 and 11. The mode in set C are 25,
37 and 45 since these numbers have the highest frequency. Each element in set D has the same
number of occurrences, thus, the data set has no mode.
The distribution of data may be classified as unimodal, bimodal, trimodal or multimodal
distribution depending upon the number of modal values in the given data set. In the above
example, set A is unimodal, set B is bimodal and set C is trimodal.
Example 2:
What is the modal color of the shirt worn by the students if the data gathered were as follows:
white, gray, gray, black, white, red, red, gray, black, white, white, red, gray, red, gray, black, red,
red, gray, gray, black?
Solution:
Since gray has the highest frequency, it follows that the modal color of the shirt worn by the
students is gray.
Remarks: 1. The mode can be used for both quantitative and qualitative data. 2. It is very much
affected by the method of grouping. 3. It is determined by the frequency and not by the values
of the observations.
MEASURE OF DISPERSION
In some cases, describing the data using the measures of central tendency alone is not enough
to provide a sufficient information concerning a population or sample. It should be supplemented
by an analysis on how the individual elements of the population/sample tends to cluster around
the central tendency. Thus, an analysis on the variability of the observations may be applied.
The most commonly used measures of dispersion are the range, variance, and standard
deviation. The simplest measure and easiest to compute but a rough estimate for the measure
of dispersion is the range.
A measure of dispersion/measure of variation is a quantity that measures the spread or variability
of the values in a given set of data.
The range, R, is the difference between the highest value (H) and lowest value (L) in the data set.
That is, R = H – L.
In terms of measure of central tendency, each student performs equally since they have same
average rating of 80%. However, looking at the variability of their ratings, Student A has the
highest range as compared to the other students. This shows that scores of student A are more
dispersed than the other. The rating of Student A is fluctuating while that of Student B is
uniformly distributed. On the other hand, Student C has range equal to zero so his ratings are all
concentrated at its mean indicating that the distribution has no spread.
Example 2.
The average daily allowances (in pesos) of 12 college students studying at University Y are 112,
127, 118, 147.5, 165.5, 99.75, 150, 145, 145, 102, 136.25 and 113. Find the range.
Solution:
Remarks:
1. The larger the value of the range, the more dispersed the observations are.
2. The range considers only the extreme values or observations in the data set.
A more reliable measure in describing the spread of a set of observations is the standard
deviation. Most researches uses this measure in the treatment of data. The computation includes
all the values in the data set.
The standard deviation is the positive square root of the variance. The variance is the average of
the squared deviations of every observation from the mean
The standard deviation and variance can be obtained from a population and a sample but most
its applications utilizes the sample rather than the population due to the complete enumeration
of the latter. The unit of the variance is squared unit while that of the standard deviation is the
same as the unit of the data set. The following symbols are used to designate these measures to
a population and sample.
The variance and standard deviation of a population are calculated by using the formulas below.
Variance and Standard deviation of Population: Consider
a population. Then, the population variance is
deviation is
be the N elements of
and the population standard
.
Sample Variance: Let
be the random sample of observations. Then, the sample
variance is
and the standard deviation of the sample is
.
Example 1: The following are the scores of a student in all her long exams in Calculus: 83, 80, 89,
78, and 70. Calculate the standard deviation.
Solution:
78
-2
4
The result indicates that on the average, the percentage scores of the student tends to deviate
from the mean by an amount of 6.23 units.
Example 2: The following data were obtained by sampling on a population. 10 12 14 15 17 18 18
24 Find the variance and the standard deviation of the sample.
Solution:
Total
10
12
14
15
17
18
18
24
128
-6
-4
-2
-1
1
2
2
8
36
16
4
1
1
4
4
64
130
The variance is 18.57 while the standard deviation is approximately 4.31. What can you infer from
this? Remarks: A large amount of standard deviation indicates that, on the average, the data
values will be far from the mean while the standard deviation of smaller amount shows that, on
the average, the data values will be close to the mean.