Mean, median and mode are some of the measures of central tendency.
Mean :
The arithmetic mean of a given data is the sum of all observations divided by the number of observations.
Median :
The median is nothing but the middle – or “mid” – of all the values presented in the data set.
There may be odd or even number of data values.
If n is odd, then using the formula
Median = (n+1)^{th} term/2
If n is even, then using the formula
Median = [(n/2)^{th} term + (n/2+1)^{th} term]/2
Mode :
The observation with the highest frequency is called mode.
Range :
The difference between the largest value and the smallest value is known as range.
Find the mean, median, mode, and range for the data set:
Problem 1 :
1, 3, 4, 5, 9, 9, 11
Solution :
(i) Mean
By using the mean formula,
Mean = sum of all observations/number of observations
Here, n = 7
= (1 + 3 + 4 + 5 + 9 + 9 + 11)/7
= 42/7
= 6
Mean = 6
(ii) Median
Given data, 1, 3, 4, 5, 9, 9, and 11 in ascending order
So, Median = middle value
Median = 5
(iii) Mode
Given data, 1, 3, 4, 5, 9, 9, 11
In the data, 9 occur the most often value.
So, Mode = 9
(iv) Range
Range = Large value - small value
= 11 - 1
= 10
Range = 10
Problem 2 :
10, 12, 12, 15, 15, 17, 18, 18, 18, 19
Solution :
(i) Mean
By using the mean formula,
Mean = sum of all observations/number of observations
Here, n = 10
= (10 + 12 + 12 + 15 + 15 + 17 + 18 + 18 + 18 + 19)/10
= 154/10
= 15.4
Mean = 15.4
(ii) Median
Given data, 10, 12, 12, 15, 15, 17, 18, 18, 18, 19 in ascending order
Here, n = 10(even)
By using the median formula,
Median = [(n^{th}/2) term + (n/2+1)^{th} term]/2
= (5^{th} term + 6^{th} term)/2
= (15 + 17)/2
= 32/2
= 16
Median = 16
(iii) Mode
Given data, 10, 12, 12, 15, 15, 17, 18, 18, 18, 19
In the data, 18 occur the most often value.
So, Mode = 18
(iv) Range
Range = Large value - small value
= 19 - 10
= 9
Range = 9
Problem 3 :
8, 4, 17, 11, 10, 10, 12, 11, 9, 18, 11, 6, 17, 7, 8
Solution :
(i) Mean
By using the mean formula,
Mean = sum of all observations/number of observations
Here, n = 15
= (4 + 6 + 7 + 8 + 8 + 9 + 10 + 10 + 11 + 11 + 11 + 12 + 17 + 17 + 18)/15
= 159/15
= 10.6
Mean = 10.6
(ii) Median
Given data, 4, 6, 7, 8, 8, 9, 10, 10, 11, 11, 11, 12, 17, 17, 18 in ascending order
So, Median = middle value
Median = 10
(iii) Mode
Given data, 4, 6, 7, 8, 8, 9, 10, 10, 11, 11, 11, 12, 17, 17, 18
In the data, 11 occur the most often value.
So, Mode = 11
(iv) Range
Range = Large value - small value
= 18 - 4
= 14
Range = 14
Problem 4:
127, 123, 115, 105, 145, 133, 142, 115, 135, 148, 129, 127, 103, 130, 146, 140, 125, 124, 119, 128, 141, 116
Solution :
(i) Mean
By using the mean formula,
Mean = sum of all observations/number of observations
Here, n = 22
= (103 + 105 + 115 + 115 + 116 + 119 + 123 + 124 + 125 + 127 + 127 + 128 + 129 + 130 + 133 + 135 + 140 + 141 + 142 + 145 + 146 + 148)/22
= 2816/22
= 128
Mean = 128
(ii) Median
Given data, 103, 105, 115, 115, 116, 119, 123, 124, 125, 127, 127, 128, 129, 130, 133, 135, 140, 141, 142, 145, 146, 148 in ascending order
Here, n = 22(even)
By using the median formula,
Median = [(n^{th}/2) term + (n/2+1)^{th} term]/2
= (11^{th} term + 12^{th} term)/2
= (127 + 128)/2
= 255/2
= 127.5
Median = 127.5
(i) Mode
Given data,
103, 105, 115, 115, 116, 119, 123, 124, 125, 127, 127, 128, 129, 130, 133, 135, 140, 141, 142, 145, 146, 148
In the data, 115, 127 occur the most often values.
So, Mode = 115, 127
(ii) Range
Range = Large value - small value
= 148 - 103
= 45
Range = 45
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