By: David Negrelli
The ability to describe data in various ways
has always been important. The need to
organize masses of information has led to the
development of formalized ways of describing
data. The purpose of this presentation is to
introduce the reader to basic tenants of
statistics.
 Frequency distribution
 Measures of central tendency
 Measures of dispersion
 Statistical significance
 T-test calculations
 Degrees of freedom
 Levels of significance
 Finding the critical value of T
 Filling out summary table
 Writing the results
Data is collected and often organized into
formats that are interpreted easily.
Example: Plant height due to the application of fertilizers.
Height is given in centimeters (cm.)
10
15
12
13
9
14 11
12 13
8 12
11 12
16
7
12
14
9
8
11
15
13
10
10
9
Number of plants
10
15
12
13
9
7
12
12
11 12 13
11 12 13 14 15
8
9
9
10
10
8
9
10 11 12 13 14 15
Height (cm)
16
14
12
8
11
16
11 12 15
13 14 13
12
9 10
12
8 10
7
11 9
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
Median- The middle number in a set of data.
Mode- The number within the set of data that
appears the most frequently.
Mean- The average
a. Denoted by х
b. Calculated by the following
formula Х = Σx
n
 Variance- Determined by averaging the squared
difference of all the values from the mean.
- symbolized by δ2
δ2 = Σ (х – х)2
n-1
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Standard Deviation- Is a measure of
dispersion that defines how an individual
entry differs from the mean.
- calculated by finding the square root of the
variance. δ = √ δ2
Defines the shape of the normal distribution
curve
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The red area represents the first standard deviant.
68% of the data falls within this area.
Calculated by x ± δ
The green area represents the second standard deviant.
95% of the data falls within the green PLUS the red area.
Calculated by x ± 2δ
The blue area represents the third standard deviant.
99% of the data falls within blue PLUS the green PLUS the red area.
Calculated by x ± 3δ
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Statistical significance is calculated by
determining:
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if the probability differences between sets of data
occurred by chance
or were the result of the experimental treatment.
Two hypotheses need to be formed:


Research hypothesis- the one being tested by the
researcher.
Null hypothesis- the one that assumes that any
differences within the set of data is due to chance
and is not significant.

Example of Null Hypothesis:
The mean weight of college football
players is not significantly different
from professional football players.
µcf = µpf
µ, ‘mu’ symbol for
Null Hypothesis
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Statistical test that helps to show if there is a
real difference between different treatments
being tested in a controlled scientific trial.
The Student t test is used to determine if the
two sets of data from a sample are really
different?

The uncorrelated t test is used when no
relationship exist between measurements in the
two groups.
Two basic formulas for calculating an uncorrelated t
test.
Equal sample size

t=
x1 – x2
n
√
Unequal sample size
δ21 + δ22
t=
√
x1 – x2
( n1 – 1)δ21 + ( n2 – 1) δ22
n1 + n2 – 2
∙(
1 +1
n1 n 2
)
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Represents the number of independent
observations in a sample.
Is a measure that states the number of
variables that can change within a statistical
test.
Calculated by n-1 ( sample size – 1)
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Is determined by the researcher.
Symbolized by α
Is affected by the sample size and the nature of the
experiment.
Common levels of significance are
.05, .01, .001
Indicates probability that the researcher made an
error in rejecting the null hypothesis.
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A probability table is used
First determine degrees of freedom
Decide the level of significance

Example: degrees of freedom= 4
α= .05
 The critical value of t= 2.776
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If the calculated value of t is less than the
critical value of t obtained from the table, the
null hypothesis is not rejected.
If the calculated value of t is greater than the
critical value of t from the table, the null
hypothesis is rejected.

The following information is needed in a summary
table
Descriptive statistics
Mean
Variance
Standard deviation
1SD (68% Band)
2 SD (95% Band)
3 SD (99% Band)
Number
Results of t test

Example: Data obtained from a experiment
comparing the number of un-popped seeds in
popcorn brand A and popcorn brand B.
A
B
26
32
22
35
30
20
34
33
Is the difference significant?

Determine mean, variance and standard deviation
of samples.
Mean xA = Σx
= 26+22+30+34
n
= 23
4
Mean xB = Σx
n
= 32+35+20+33
= 30
4
δ2= Σ (х – х)2
n-1
variance
Popcorn A = ( 26-23)2 + (22-23)2 + (30-23)2 + (34-23)2
3
= 9 + 1 + 49 + 121
3
= 60
Popcorn B = ( 30-30)2+ (35-30)2 + (20- 30)2 + (33- 30)2
3
= 0 + 25 + 100 + 9
3
= 44.67
Standard deviation: δ= √ δ2
popcorn A
√ 60
= 7.75
Popcorn B
√ 44.67 = 6.68
x1 – x2
t=
Finding Calculated t
√
δ21 + δ22
n
t = 23 - 30
√
=
60+ 44.67
4
7
√ 26.17
=
7
5.12
= 1.38
Determine critical value of t
• Select level of significance
α=.01
• Determine degrees of freedom
degrees of freedom of A= 3
degrees of freedom of B= 3
total degrees of freedom = 6
• Critical value of t = 3.707
Calculated value of t =1.38 is less than critical value
of t from the table, 3.707.
The null hypothesis is not rejected.
Descriptive statistics
popcorn A
popcorn B
Mean
23
60
30
44.67
7.75
15.25 - 30.75
7.50-38.50
-.25 - 46.25
4
6.68
23.32- 36.68
16.64-43.36
9.96-50.04
4
Variance
Standard deviation
1SD (68% Band)
2 SD (95% Band)
3 SD (99% Band)
Number
Results of t test
t= 1.38
t of 1.38 < 3.707
df=6
α=.01
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Write a topic sentence stating the independent and
dependent variables and a reference to a table or
graph.
Write sentences comparing the measures of central
tendency of the groups.
Write sentences describing the statistical tests,
levels of significance, and the null hypothesis.
Write sentences comparing the calculated value with
the required statistical value. Make a statement about
rejection of the null hypothesis.
Write a sentence stating support of the research
hypothesis by the data.