Measure of
Central Tendency
It indicates where the center of the distribution tends
to be located. It refers to the typical or average score
in a distribution.
a. Mean
▪ It is the exact
mathematical center
of a distribution. It is
equal to the sum of all
scores divided by the
number of cases or
observations.
▪Example 1:
3 + 8 + 9 + 2 + 3 + 10 + 3
= 38
38 ÷ 7
= 5.43 (Mean)
Example 2:
Mean (𝑋 )=
̅ 1962 ÷ 20 = 98.1 (Mean)
Weighted
Mean
▪ The weighted mean is necessary in
some situations. Suppose that you
are given the means of two or more
measurements and you wish to find
the mean of all the measures
combined into one group. The
formula for weighted mean is given
by.
where:
f = frequency
x = numerical value or
item in a set of data
n = number of
observations in the data set
Example: Find the mean of the heights of 50 senior
high school students summarized as follows:
Heights (in inches) Frequency Heights x
Frequency
56 6
336
57 15
58 12 855
59 8 696
60 5 472
61 2 300
62 2 122
= =
124
50 2905
Solution: Using the above data, the weighted mean is equal
to the sum of the column fx, divided by the total number
of observation.
= 2905 ÷ 50
= 58.1 inches
b. Median
▪ it is the midpoint of the distribution. It
represents the point in the data where 50% of
the values fall below that point and 50% fall
above it. When the distribution has an even
number of observations, the median is the
average of the two middle scores. The median
is the most appropriate measures of central
tendency for ordinal data.
The median may be calculated from ungrouped
data by doing the following steps:
1. Arrange the items (scores, responses,
observations) from lowest to highest.
2. .2. Count to the middle value, for an odd
number of values arranged from lowest to
highest, the median corresponds to value, if
the array contains an even number of
observations, the median is the average of
the two middle values.
Example 1: Consider these odd numbers
of numerical values:
7,8,8,9,10,12,23
The median is 9.
Example 2: Consider these even number
of numerical values:
12, 15, 18, 22, 30, 32
18 + 22 = 40
40 ÷ 2 = 20
The median is 20
c. Mode
▪ It refers to the most frequently occurring score in a
distribution. In cases where there is more than one
observation which is the highest but with equal
frequency, the distribution is bimodal (with 2
highest observations) or multimodal with more
than two highest observations. In cases where
every item has an equal number of observations,
there is no mode. The mode is appropriate for
nominal data.
Example 1: The ages of fifteen (15) persons
assembled in a room are as follows:
16, 18, 18, 25, 25, 25, 30, 34, 36 and 38.
Solution: An age of 25 is the mode
because it has been recorded three
times in the sample, more than any
other age.
Example 2: The number of hours spent by 10 students
in an internet café was as follows:
2, 2, 2, 3, 3, 4, 4, 4, 5, 5
Solution: Both 2 and 4 have a frequency
of 3. The data is therefore bimodal.
Answer: Mode = 2 and 4
Measures of dispersion
1. Range – it is the difference between the
largest and the smallest values in a set of data.
Consider the following scores obtained by
ten (10) students participating in a
mathematics contest:
6, 10, 12, 15, 18, 18, 20, 23, 25, 28 = (28-6=22)
Thus, the range is 22. The scores range from 6
to 28
2. Average (Mean) Deviation
This measure of spread is defined as the absolute
difference or deviation between the values in a set of
data and the mean, divided by the total number of
values in the set of data.
In mathematics, the term “absolute” represented
by the sign “ l l” simply means taking the value of a
number without regard to positive or negative sign.
▪ Example 1: Consider a set of values which consists
of 20, 25, 35, 40, and 45. Solving for the mean,
20+25+35+40+45
5
= 33
- |- |
20-33 -13 13
25-33 -8 8
35-33 2 2
40-33 7 7
45-33 12 12
= 33 = 8.4
Standard
Deviation
▪ The standard deviation (SD) is a
measure of the spread or variation
of the data about the mean.
▪ The formula for calculating the standard
deviation is given by
▪ Example: Let us consider the
same data used in the
illustration for using the range.
The values are 6, 10, 12, 15, 18,
18, 20, 23, 25, and 28.
▪ Solution:
1. Compute the mean
= 6 + 10 + 12 + 15 + 18 + 18 + 20 + 23 + 25 + 28
10
= 175/ 10
= 17.5
2. Subtract the mean ̅ from each score
(), or - .̅
3. Square each difference from step 2,
or ( - )2
̅
Scores () (- ) (x-x)2
6 (6-17.5)=-11.5 132.25
10 (10-17.5) = -7.5 56.25
12 (12-17.5) = -5.5 30.25
15 (15-17.5) = -2.5 6.25
18 (18-17.5) = .5 .25
18 (18-17.5) = .5 .25
20 (20-17.5) = 2.5 6.25
23 (23-17.5) = 5.5 30.25
25 (25-17.5) = 7.5 56.25
28 (28-17.5) = 10.5 110.25
Σ (x - x𝑥̅)2
= 428.5
4. Sum all the squares from step 3 or Σ ( - )2
̅
5. Divide the number in step 4 by n-1. The
number of items or scores is denoted by n. The
quality n-1 is called the degrees of freedom, a
statistical concept that procedures a more
accurate estimate of the data.
▪ 6. Compute the standard deviation using the formula
below.
= =
=
SD =6.9
Testing the
hypothesis
Statistical Significance
▪ Statistically significant means that a
relationship between two or more
variables is caused by something other
than by random chance. Significant also
means probably true (not due to chance).
When the result is highly significant, it
means that it is very probably true.
The level of significance shows how likely the
results of your data are due to chance. A chance
of being true indicates that the finding has a
five percent chance of not being true. A level of
significance means that there is a chance that
the finding is true.
Statistical hypothesis testing is used to
determine whether the result of a data set is
statistically significant.
Hypothesis
▪ A hypothesis is a preconceived idea, assumed to be
true and has to be tested for its truth or falsity. Let
us suppose a researcher is concerned with testing
the relationship between variables. Through
inferential statistical measures, the researcher can
discover important information even if no
relationship is established between the variables. It
is possible for the researcher to discover differences
and, therefore may test individual or group
differences.
Type I and Type II Errors
▪ There are two types of errors involved with
hypothesis testing. Type I error is committed
when a researcher rejected a null hypothesis
when in fact it is true. The second type or
error. Type II error, is the error that occurs
when the data from the sample produce
results that fail to reject the null hypothesis
when in fact the null hypothesis is false and
should be rejected.
Parametric and Nonparametric
Statistics
▪ Two types of significance tests that are
usually used by researchers are called
parametric and nonparametric statistics.
Parametric tests are used for interval and
ratio scales of measurement. They require
that the samples and observation are drawn
from normally distributed populations and
that the selection of each case should be
independent of the other. The population
should have equal variances.
Nonparametric tests do not specify
normally distributed populations and
similarity of variances. Nonparametric
tests are the only tests used with
nominal data or ordinal data.
Steps in Hypothesis Testing
The main concern or idea in hypothesis
testing is to ensure that what is observed
from sample data and generalized to
population phenomena is not due to
chance.
The following outlines the steps in
hypothesis testing in any given situation.
1. State the null hypothesis. The null hypothesis is a
statement that no difference exist between the
averages or means of two groups.
2. Choose the statistical test and perform the
calculation. A researcher must determine the
measurement scale, the type of variable, the type of
data gathered and the number of groups or the
number of categories.
3. State the level of significance for the statistical test.
The level of significance is determine before the test is
performed.
4. Compute the calculated value. Use the appropriate
formula (discussed in the previous lesson) for the
significance test to obtain the calculated value.
5. Determine the critical value the test statistic must
attain to be significant.
6. Make the decision. If the calculated value is greater
than the critical value, you reject the null hypothesis.
If the critical value is larger, you conclude that you
have failed to reject the null hypothesis.
