MODULE
10
Measures of Central Tendency
Learning Competencies:
i.
illustrates the measures of central
tendency (mean, median, and mode)
of a statistical data. M7SP-IVf-g-1
ii.
calculates the measures of central
tendency of ungrouped and grouped
data. M7SP-IVh-1
Median
The element in the middle of the data
set after it’s arranged in order.
Example 1:
34,68,84,76,61,61,85,54,44,70,72,83,61
Arrange from lowest to highest.
34,44,54,61,61,61,68,70,72,76,83,84,85
Measure of Central Tendency
(Mean, Median, Mode)
Statistics – collecting and understanding
data.
Measure of Central Tendency (also
referred to as measures of centre or central
location) is a summary measure that attempts
to describe a whole set of data with a single
value that represents the middle or centre of
distribution.
Solution:
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠
=
2
=
13
2
= 6.5 (start count from lowest to highest)
34,44,54,61,61,61, 68 ,70,72,76,83,84,85
There are three main measures of
central tendency: the mean, the median and
the mode.
Each of these measures describes a
different indication of the typical or central
value in the distribution.
68 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑑𝑖𝑎𝑛.
Example 2:
61,59,34,40,60,95,73,45
Arrange from lowest to highest.
Mean
The sum of all the numbers in the data
set divided by the number of elements.
Example:
34,68,84,76,61,61,85,54,44,70,72,83,61
34,40,45,59,60,61,73,95
Solution:
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠
=
2
=
8
2
= 4 (start count from lowest to highest)
Solution:
34 + 68 + 84 + 16 + 61 + 61 + 85 + 54 + 44 + 70 + 72 + 83 + 61
=
13
=
853
13
= 65.62
34,40,45, 59 , 60 ,61,73,95
59 + 60 119
=
= 59 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑑𝑖𝑎𝑛.
2
2
Mode
Answer: 61 and 54
The values that occurs frequently in the
data set.
34,44,54,54,61,61,68,70,72,76,83,84,85
Example 1:
34,68,84,76,61,61,85,54,44,70,72,83,61
Arrange from lowest to highest.
34,44,54,61,61,61,68,70,72,76,83,84,85
Answer: 61
Try Me!
Direction: Find the mean, median and mode.
1.) 89, 75, 53, 39,
89, 98, 74, 11,
18, 15, 34, 77
2.)23, 37, 11, 58, 13, 45
85, 42, 64, 71, 39, 56
53, 22, 76, 46, 68, 32
60, 52, 11, 29, 46, 9
34,44,54,61,61,61,68,70,72,76,83,84,85
Example 2:
61,59,34,40,40,95,73,45
Arrange from lowest to highest.
34,40,40,45,59,61,73,95
Answer: 40
34,40,40,45,59,61,73,95
Example 3:
61,59,34,40,60,95,73,45
Arrange from lowest to highest.
34,40,45,59,60,61,73,95
Measure of Central Tendency
(Grouped and Ungrouped Data)
Measures of Central Tendency or
Average occurs regularly in our daily life and it
is an important tool in statistics. A well-chosen
average consists of a single number about
which a given data are centered. There can be
different types of averages or sometimes called
measures of central tendency. They are the
mean, median and mode.
Mean
The most used measure of central
tendency. When we speak of average, we
always refer to the mean.
It is found by adding the values of the
data and dividing by the total number of
values.
Answer: None
Population Mean
𝝁=
∑𝒙
𝑵
̅=
𝒙
∑𝒙
𝒏
Example 4:
34,68,84,76,61,61,85,54,44,70,72,83,54
Arrange from lowest to highest.
34,44,54,54,61,61,68,70,72,76,83,84,85
Sample Mean
Where 𝑵 is the total number of observations
in the population. 𝒏 is the total number of
observations in the sample.
Example 1 (Ungrouped Data)
A researcher collect data on the ages of
the recipients of doctoral degree in science and
engineering, and his study yields the following:
37, 37, 24, 28, 43, 44, 36, 41, 33, 27
Solution:
∑𝒙
̅=
𝒙
𝒏
̅=
𝒙
37 + 37 + 24 + 28 + 43 + 44 + 36 + 41 + 33 + 27
10
̅=
𝒙
350
10
𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =
𝑙𝑜𝑤𝑒𝑟 𝑐𝑙𝑎𝑠𝑠 + 𝑢𝑝𝑝𝑒𝑟 𝑐𝑙𝑎𝑠𝑠
2
𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 =
Class
Boundaries
52.5 – 63.5
63.5 – 74.5
74.5 – 85.5
85.5 – 96.5
96.5 – 107.5
107.5 – 118.5
52.5 + 63.5
= 58
2
Frequency
Midpoint(𝒙𝒎 )
6
12
25
18
14
5
58
69
80
91
102
113
Step 2. Multiply the frequency and the
midpoint in each class boundary.
̅ = 𝟑𝟓
𝒙
For grouped data, the midpoint of the
classes is used for the values of the x.
The following are the steps in solving for
the mean of grouped data.
Grouped Mean
1. Find the midpoint for each class. Place
them in a column.
2. Multiply the frequency by the midpoint
for each class. Place them in another
column.
3. Find the sum of the resulting column in
step 2 and step 4.
4. Divide the sum obtained in step 3 by the
total number of frequencies. That is,
𝑚𝑒𝑎𝑛 =
∑ 𝑓 ∗ 𝑥𝑚
𝑛
Example 2 (Grouped Data)
Eighty randomly selected light bulbs
were tested to determine their lifetime (in
hours). The following frequency was obtained.
Find the mean.
Class Boundaries
Frequency
52.5 – 63.5
63.5 – 74.5
74.5 – 85.5
85.5 – 96.5
96.5 – 107.5
107.5 – 118.5
6
12
25
18
14
5
Step 1. Solve for the midpoint in each class
boundary.
Class
Frequency Midpoint
𝒇 ∗ 𝒙𝒎
(𝒇)
(𝒙𝒎 )
Boundaries
52.5–63.5
6
58
348
63.5–74.5
12
69
828
74.5–85.5
25
80
2000
85.5 – 96.5
18
91
1638
96.5–107.5
14
102
1428
107.5-118.5
5
113
565
Step 3. Find the sum of the 2nd and 4th
Column.
Frequency Midpoint
Class
𝒇 ∗ 𝒙𝒎
(𝒇)
(𝒙𝒎 )
Boundaries
52.5–63.5
6
58
348
63.5–74.5
12
69
828
74.5–85.5
25
80
2000
85.5 – 96.5
18
91
1638
96.5–107.5
14
102
1428
107.5-118.5
5
113
565
80
6807
𝑚𝑒𝑎𝑛 =
∑ 𝑓 ∗ 𝑥𝑚
𝑛
𝑚𝑒𝑎𝑛 =
6807
80
𝑚𝑒𝑎𝑛 = 85.09
Median
The median is the midpoint of the data
array. Before finding its value, the data must
be arranged in order, from least to greatest or
vice versa. The median will either be a specific
value or will fall between two values.
If there is an odd number of numbers,
the median value is the middle most number,
with the same number of numbers below and
above.
Example 3 (Ungrouped Data)
Seven mothers were selected and given
a blood pressure check. Their systolic pressure
was recorded below.
Step 1. Make a table of cumulative frequency.
Time (in
seconds)
Frequency
<cf
51-55
56-60
61-65
66-70
2
7
8
4
2
9
17
21
Step 2. Divide n, number of frequencies by 2,
to get the halfway point.
135, 121, 119, 116, 130, 121, 131
𝑁 21
=
= 10.5
2
2
Find their median.
Arrange from least to greatest or vice versa.
116, 119, 121, 121, 130, 131, 135
Grouped Median
1. Make a table of cumulative frequency.
2. Divide n, number of frequencies by 2, to
get the halfway point.
3. Locate the median class in the
cumulative frequency column.
4. Substitute in the formula,
Step 3. Locate the median class in the
cumulative frequency column.
Time (in
seconds)
Frequency
<cf
51-55
56-60
61-65
66-70
2
7
8
4
2
9
17
21
Step 4. Substitute in the formula,
𝑛
− 𝑐𝑓
) (𝑤) + 𝐿𝑚𝑑
𝑚𝑒𝑑𝑖𝑎𝑛 (𝑀𝑑 ) = (2
𝑓
𝑛
− 𝑐𝑓
) (𝑤) + 𝐿𝑚𝑑
𝑚𝑒𝑑𝑖𝑎𝑛 (𝑀𝑑 ) = (2
𝑓
Where,
n – sum of frequencies
cf – cumulative frequency of the class
preceeding/before the median class
f – frequency of the median class
w – class width
21
−9
𝑚𝑒𝑑𝑖𝑎𝑛 (𝑀𝑑) = ( 2
) (5) + (60.5)
8
𝑚𝑒𝑑𝑖𝑎𝑛 (𝑀𝑑) = (
𝐿𝑚𝑑 – lower boundary of the median class
Example 4 (Grouped Data)
The record of 21 people in a 100m race
is summarized in the given frequency table.
Determine the median of the given data.
Time (in seconds)
Frequency
51-55
56-60
61-65
66-70
2
7
8
4
10.5 − 9
) (5) + (60.5)
8
= 61.44
Mode
The third measure of average is
the mode. It is the value that occurs
most often in the data set. A data can
have more than one or none at all. The
mode for grouped data is the modal
class. The modal class is the class with
the largest frequency. The mode is the
only measure of central tendency that
can be used in finding the most typical
case when the data are nominal or
categorical.
Mode (Ungrouped Data)
To find the mode of ungrouped
data, find the frequency of each
number/ value/ observation in the given
data set. Then choose the number/
value/ observation having the highest
frequency as the mode.
𝑑1
𝑚𝑜𝑑𝑒 (𝑀𝑜) = 𝐿𝑚𝑜 + (
) (𝑤)
𝑑1 + 𝑑2
Where,
𝐿𝑚𝑜 – lower boundary of modal class
W – class width
𝑑1 – difference of the frequency of the
modal class and the class preceeding it.
𝑑2 - difference of the frequency of the
modal class and the class succeeding it.
Example 5 (Ungrouped Data)
Find the mode of the given data set:
Step 1. Identify the modal class by
determining the interval with the highest
frequency.
Time (in seconds)
Frequency
51-55
56-60
61-65
66-70
2
7
8
4
Step 2. Determine the exact lower limit of
the modal class.
Time (in seconds)
Frequency
51-55
56-60
61-65
66-70
2
7
8
4
60.5
Step 3. Substitute in the formula.
15,28,25,48,
22,43,39,44,
43,49,34,22,
33,27,25,22,30
𝑑1
𝑚𝑜𝑑𝑒 (𝑀𝑜) = 𝐿𝑚𝑜 + (
) (𝑤)
𝑑1 + 𝑑2
Solution:
First arrange the data set in ascending
or descending order.
1
𝑚𝑜𝑑𝑒 (𝑀𝑜) = 60.5 + (
) (5)
1+4
𝑚𝑜𝑑𝑒 (𝑀𝑜) = 61.5
15,22,22,22,
25,25,27,28,
30,33,34,39,
43,43,44,48,49
In the given data set, the number that
appeared the most number of times is 22.
Example 6 (Grouped Data)
The record of 21 people in a 100m race
is summarized in the given frequency table.
Determine the mode of the given data.
Time (in seconds)
Frequency
51-55
56-60
61-65
66-70
2
7
8
4
Try this!
Direction: Find the mean, median, and
mode of Grouped Data.
A mathematician achievement test
contained 40 questions for which the answers
were marked either right or wrong. The
distribution below summarize the results. Find
the mean, median and the mode of grouped
data.
No. of Answers
Correct
0-2
3-5
6-8
9-11
12-14
15-17
18-20
21-23
24-26
27-29
30-32
33-35
36-40
Frequency
Prepared by:
JUANITO JR D. BERGADO
Mathematics Teacher
0
1
3
10
11
17
24
25
22
16
7
3
0
Approved by:
LUZMINDO A. PAESTE
School Principal