The document discusses measures of central tendency, including the mean, median, and mode. It provides an example of monthly family incomes and calculates each measure using the data. The mean is the average, the median is the middle value, and the mode is the most frequent value. Each measure has unique properties that determine when it is most appropriate to use.

1. Measures of Central
Tendency
Samuel John E. Parreño
Special Science Teacher 1

2. Learning Outcomes
Calculate commonly used measures of
central tendency,
Provide a sound interpretation of these
summary measures, and
Discuss the properties of these measures.

3. Monthly Family Income in
Pesos
Number of
Families
12,000 2
20,000 3
24,000 4
25,000 8
32,250 9
36,000 5
40,000 2
60,000 2

4. 1. What is the highest monthly family income?
Lowest?
Answer: Highest monthly family income is 60,000 pesos while
the lowest is 12,000 pesos.
The highest and lowest values, which are commonly known
as maximum and minimum, respectively are summary
measures of a data set. They represent important location
values in the distribution of the data. However, these
measures do not give a measure of location in the center of
the distribution.

5. 2. What monthly family income is most frequent in
the village?
Answer: Monthly family income that is most frequent is 32,250
pesos.
The value of 32,250 occurs most often or it is the value with
the highest frequency. This is called the modal value or
simply the mode. In this data set, the value of 32,250 is found
in the center of the distribution.

6. 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?
Answer: When arranged in increasing order or the data come
in an array as in the following:
12,000; 12,000; 20,000; 20,000; 20,000; 24,000; 24,000;
24,000; 24,000; 25,000; 25,000;25,000; 25,000; 25,000;
25,000; 25,000; 25,000; 32,250; 32,250; 32,250; 32,250;
32,250; 32,250; 32,250; 32,250; 32,250; 36,000; 36,000;
36,000; 36,000; 36,000; 40,000; 40,000; 60,000; 60,000;

7. 4. What is the average monthly family income?
Answer: When computed using the data values, the average is
30,007.14 pesos.
The average monthly family income is commonly referred to
as the arithmetic mean or simply the mean which is
computed by adding all the values and then the sum is
divided by the number of values included in the sum. The
average value is also found somewhere in the center of the
distribution.

8. Common Measures of
Central Tendency
Mean, Median, and Mode

9. 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 𝑖=1
𝑁
𝑥𝑖 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 = 𝑖=1
𝑁
𝑥𝑖 /𝑁. 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

10. MEAN
Alternatively, we could do the computation as follows:

11. MEDIAN
The 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.

12. MEDIAN
If 𝑁 (total number of observations in a data set) is odd, the median
or the middle value is the
𝑁+1
2
𝑡ℎ
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
𝑁
2
𝑡ℎ
and
𝑁
2
+ 1
𝑡ℎ
observations.

13. MEDIAN
In the example given earlier, there are 35 observations so 𝑁 is 35,
an odd number. The median is then the
𝑁+1
2
𝑡ℎ
=
36
2
𝑡ℎ
= 18𝑡ℎ
observation in the array. Locating the 18th observation in the array
leads us to the value equal to 32,250 pesos.

14. MODE
The 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 above, the value of 32,250 pesos occurs
most often or it is the value with the highest frequency which is
equal to nine.

15. PROPERTIES OF THE
MEAN, MEDIAN AND
MODE

16. MEAN
As mentioned before 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 as
seen in the following illustration.

17. MEAN

18. MEAN
The sum of the differences across all observations will be equal
to zero. This indicate that the mean indeed is the center of the
distribution since the negative and positive deviations cancel out
and the sum is equal to zero.
In the expression given above, we could see that each
observation has a contribution to the value of the mean. All the
data contribute equally in its calculation. That is, the “weight” of
each of the data items in the array is the reciprocal of the total
number of observations in the data set, i.e. 1/𝑁.

19. MEAN
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 mean for all 30 observations
as follows:
𝜇 =
(𝑁1 𝜇1 + 𝑁2 𝜇2 + 𝑁3 𝜇3)
30
=
10 × 5 + 5 × 15 + (7 × 10)
30
=
195
30
= 6.59

20. MEAN
 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, we could tell the students that 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 + ⋯ 60,000)/35 = 2,130,250/35 = 60,864.29

21. MEDIAN
Like the mean, 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.

22. MODE
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
theoretically occur only once.
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.

24. KEY POINTS
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