This document discusses measures of central tendency including the mean, median, and mode. It defines each measure and explains their properties and appropriate uses. The mean is the average value obtained by dividing the sum of all values by the total number of values. The median is the middle value when values are arranged from lowest to highest. The mode is the most frequently occurring value. Each measure can be affected differently by outliers, with the median being least affected. The appropriate measure depends on the scale of measurement and distribution of the data.

1. MEASURES OF CENTRAL
TENDENCY

2. LEARNING OUTCOME(S): At the
end of the lesson, the learner is able
to
• Calculate commonly used measures
of central tendency.
• Provide a sound interpretation of
these summary measures, and
• Discuss the properties of these
measures.

4. Questions:
1. What is the highest monthly family income?
Lowest?
2. What monthly family income is most
frequent in the village?
3. If you list down individually the values of the
monthly family income from lowest to highest,
what is the monthly family income where half
of the total number of families have monthly
family income less than or equal to that value
while the other half have monthly family
income greater than that value?
4. What is the average monthly family income?

5. A. Common Measures of Central
Tendency: Mean, Median and
Mode
1. MEAN- the most widely used measure of
the center is the (arithmetic) mean. It is
computed as the sum of all observations in
the data set divided by the number of
observations that you include in the sum. If
we use the summation symbol, read
as ‘sum of observations represented by xi
where i takes the values from1 to N, and N
refers to the total number of observations
being added’, we could compute the mean
(usually denoted by Greek letter, µ) as µ =
/N .

6. Using the example earlier with 35
observations of family income, the
mean is computed as
µ = 12,000+12,000+⋯+60,000 /35
= 1,050,250/ 35
= 30,007.14

8. 2. MEDIAN- on the other hand is the middle
value in an array of observations. To
determine the median of a data set, the
observations must first be arranged in
increasing or decreasing order. Then
locate the middle value so that half of the
observations are less than or equal to that
value while the half of the observations
are greater than the middle value.

9. -If N (total number of observations in a data
set) is odd, the median or the middle value
is the observation in the array.
On the other hand, if N is even, then the
median or the middle value is the average
of the two middle values or it is average of
the
and observations.

10. In the example, there are 35 observations
so N is 35, an odd number. The median is
then the
= = observation in the
array.
Locating the 18th observation in the array
leads us to the value equal to 32,250
pesos.

11. 3. MODE- or the modal value is the value
that occurs most often or it is that value
that has the highest frequency. In other
words, the mode is the most fashionable
value in the data set.
Like in the example , the value of 32,250
pesos occurs most often or it is the value
with the highest frequency which is equal
to nine.

12. C. Properties of the Mean,
Median and Mode
The mean is the most commonly used
measure of central tendency since it
could be likened to a “center of
gravity” since if the values in an array
were to be put on a beam balance, the
mean acts as the balancing point
where smaller observations will
“balance” the larger ones .

14. Means are also amenable to further
computation, that is, you can combine
subgroup means to come up with the mean
for all observations. For example, if there are
3 groups with means equal to 10, 5 and 7
computed from 5, 15, and 10 observations
respectively, one can compute the mea n for
all 30 observations as follows:
µ =
=
= 195/ 30
= 6.59

15. If there are extreme large values, the mean will tend
to be ‘pulled upward’, while if there are extreme
small values, the mean will tend to be ‘pulled
downward’. The extreme low or high values are
referred to as ‘outliers’.’Thus, outliers do affect the
value of the mean.
To illustrate this property, if in case there is one
family with very high income of 600,000 pesos
monthly instead of 60,000 pesos only, the
computed value of mean will be pulled upward,
that is,
µ = 12,000+12,000+⋯+600,000 35 = 2,130,250
/35
= 60,864.29
Thus, in the presence of extreme values or outliers,
the mean is not a good measure of the center. An
alternative measure is the median. The mean is

16. The median is computed for quantitative
variables. But the median can be
computed for variables measured in at
least in the ordinal scale. Another property
of the median is that it is not easily
affected by extreme values or outliers. As
in the example above with 600,000 family
monthly income measured in pesos as
extreme value, the median remains to
same which is equal to 32,250 pesos.

17. For variables in the ordinal, the median should be
used in determining the center of the distribution.
On the other hand, the mode is usually computed
for the data set which are mainly measured in the
nominal scale of measurement. It is also
sometimes referred to as the nominal average. In
a given data set, the mode can easily be picked
out by ocular inspection, especially if the data are
not too many. In some data sets, the mode may
not be unique. The data set is said to be
unimodal if there is a unique mode, bimodal if
there are two modes, and multimodal if there are
more than two modes. For continuous data, the
mode is not very useful since here, measurements
(to the most precise significant digit) would

18. The mode is a more helpful measure for
discrete and qualitative data with
numeric codes than for other types of
data. In fact, in the case of qualitative
data with numeric codes, the mean
and median are not meaningful.

20. Generalization
• A measure of central tendency is a
location measure that pinpoints the
center or middle value.
• The three common measures of
central tendency are the mean,
median and mode.
• Each measure has its own properties
that serve as basis in determining
when to use it appropriately.