Mean :
The arithmetic mean of a given data is the sum of all observations divided by the number of observations.
Mean = (Sum of all observations)/Number of observations
Median :
The median is the middle value of a given data when those values are arranged from ascending to descending order.
Median = Middle value
To find the median from ungrouped data, we have to consider if n is odd or even.
If n is odd, then using the formula
Median = (n+1)th term/2
If n is even, then using the formula
Median = [(nth/2) term + (n/2+1)th term]/2
Mode :
The mode is the value that occurs most often in the given data.
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 = [(nth/2)term + (n/2+1)th term]/2
= (6th term + 7th 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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