FIND THE MEAN MEDIAN AND MODE OF THE GIVEN DATA SET

Example 1 :

Find the (i) mean  (ii) median  (iii) mode for each of the following data sets :

a)  12, 17, 20, 24, 25, 30, 40

b)  8, 8, 8, 10, 11, 11, 12, 12, 16, 20, 20, 24

c)  7.9, 8.5, 9.1, 9.2, 9.9, 10.0, 11.1, 11.2, 11.2, 12.6, 12.9

d)  427, 423, 415, 405, 445, 433, 442, 415, 435, 448, 429, 427, 403, 430, 446, 440, 425, 424, 419, 428, 441

Solution :

a)

(i) Mean

By using the mean formula,

Mean  =  sum of all observations/number of observations

Here, n  =  7

=  (12 + 17 + 20 + 24 + 25 + 30 + 40)/7

=  (168)/7

Mean  =  24

(ii) Median

Given data,  12, 17, 20, 24, 25, 30, 40 in ascending order.

So, Median  =  middle value

Median  =  24

(iii) Mode

Given data,  12, 17, 20, 24, 25, 30, 40

There is no mode in the given data.

b)

(i) Mean

Given data, 8, 8, 8, 10, 11, 11, 12, 12, 16, 20, 20, 24

Here, n  =  12

=  (8 + 8 + 8 + 10 + 11 + 11 + 12 + 12 + 16 + 20 + 20 + 24)/12

=  (160)/12

Mean  =  13.33

(ii) Median

Given data, 8, 8, 8, 10, 11, 11, 12, 12, 16, 20, 20, 24 in ascending order.

Here, n  =  12 (even)

By using the median formula,

Median  =  [(n^th/2)term + (n/2+1)^th term]/2

=  (6^th term + 7^th term)/2

=  (11 + 12)/2

=  (23)/2

=  11.5

Median  =  11.5

(iii) Mode

Given data, 8, 8, 8, 10, 11, 11, 12, 12, 16, 20, 20, 24

In the data, 8 occur the most often value.

So, Mode  =  8

c)

(i) Mean

Given data, 7.9, 8.5, 9.1, 9.2, 9.9, 10.0, 11.1, 11.2, 11.2, 12.6, 12.9

Here, n  =  11

=  (7.9 + 8.5 + 9.1 + 9.2 + 9.9 + 10.0 + 11.1 + 11.2 + 11.2 + 12.6 + 12.9)/11

=  (113.6)/11

Mean  =  10.32

(ii) Median

Given data, 7.9, 8.5, 9.1, 9.2, 9.9, 10.0, 11.1, 11.2, 11.2, 12.6, 12.9 in ascending order.

So, Median  =  middle value

Median  =  10.0

(iii) Mode

Given data, 7.9, 8.5, 9.1, 9.2, 9.9, 10.0, 11.1, 11.2, 11.2, 12.6, 12.9

In the data, 11.2 occur the most often value.

So, Mode  =  11.2

d)

(i) Mean

Given data, 427, 423, 415, 405, 445, 433, 442, 415, 435, 448, 429, 427, 403, 430, 446, 440, 425, 424, 419, 428, 441

Here, n  =  21

=  (427 + 423 + 415 + 405 + 445 + 433 + 442 + 415 + 435 + 448 + 429 + 427 + 403 + 430 + 446 + 440 + 425 + 424 + 419 + 428 + 441)/21

=  (9000)/21

Mean  =  428.57

(ii) Median

Given data,

427, 423, 415, 405, 445, 433, 442, 415, 435,

448, 429, 427, 403, 430, 446, 440, 425, 424, 419, 428,

441

Let’s arrange this data in ascending order :

403, 405, 415, 415, 419, 423, 424, 425, 427, 427, 428,

429, 430, 433, 435, 440, 441, 442, 445, 446, 448

So, Median  =  middle value

Median  =  428

(iii) Mode

Given data, 427, 423, 415, 405, 445, 433, 442, 415, 435, 448, 429, 427, 403, 430, 446, 440, 425, 424, 419, 428, 441

In the data, 415 and 427 are values that occur the most often.

So, Mode  =  415 and 427

Example 2  :

Consider the following two data sets :

Data set A : 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 12

Data set B : 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 20

a)  Find the mean for both Data set A and Data set B.

b)  Find the median of both Data set A and Data set B.

c)  Explain why the mean of Data set A is less than the mean of Data set B.

d)  Explain why the median of Data set A is the same as the median of Data set B

Solution :

a)

Given,  Data set A : 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 12

Mean  =  (5 + 6 + 6 + 7 + 7 + 7 + 8 + 8 + 9 + 10 + 12)/11

=  (85)/11

Mean  =  7.727

Set A : 7.73

Data set B : 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 20

Mean  =  (5 + 6 + 6 + 7 + 7 + 7 + 8 + 8 + 9 + 10 + 20)/11

=  (93)/11

Mean  =  8.45

Set B : 8.45

b)

Given,  Data set A : 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 12 in ascending order.

So, median  =  middle value

Median  =  7

Set A : 7

Data set B : 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 20 in ascending order.

So, median  =  middle value

Median  =  7

Set B : 7

c)

The data sets are the same except for the last value.

The last value of A is less than the last value of B.

So, the mean of A is less than the mean of B.

d)

The middle value of both data sets is the same.

So, the median is the same.

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