Statistics is a division of applied mathematics which contracts with the particular interpolation of data. The term ‘Statistics’ has been taken from the Latin name ‘Status’ in which it defines ‘political state’. This statistics mainly used to measure the arithmetic mean, median, mode and standard deviation. This measurement gives idea about where the data points are centered. Let us discuss about standard deviation with some example problems.
Evaluation of Standard Deviation in statistics:
Mean:
In Statistics, Mean is defined as the average of the given total numbers, i.e., total number of data divided by the number of data set given.
Formula for finding mean,
barx = (sum(x))/(n)
Standard Deviation:
In Statistics, Standard Deviation is the measure of describing squared mean difference variability and spread of the Data set in the given total numbers. It is used to take the measurement of taking square root and average of numbers in the Data set.
Formula for standard deviation,
S = sqrt ((sum(x - barx)^2 )/ (n-1))
Here Standard Deviation is calculated by using the mean Value barx
Example Problems to find standard deviation in statistics :
Problem 1:
Here are 4 measurements 66, 45, 67, 45, 34, 56, 78 and 57. Calculate statistics standard deviation for the given measurements
Sol:
Mean: Calculate the mean the using the formula,
barx = (sum(x)) / n
barx =( 66+45+67+45+34+56+78+57) / 8
= 448 / 8
barx = 56
Standard Deviation,
S = sqrt((sum(x-barx)^2) / (n-1) )
S = sqrt((( 66-56)^2+(45-56)^2+(67-56)^2+(45-56)^2+(34-56)^2+(56-56)^2+(78-56)^2+(57-56)^2) / (8-1))
= sqrt((100+121+121+121+484+0+484+1)/7)
= sqrt( 1432 / 7)
S = sqrt(204.571429)
Standard Deviation S = 14.3028469
Problem 2:
Find the Statistics Standard deviation of the given Data 16, 17, 18, 20 and 24
Sol:
Mean: Calculate the mean using the formula,
barx = (sum(x)) / n
barx = ( 16 + 17 + 18 + 20 + 24 ) / 5
barx = 95 / 5
barx = 19
X x-barx (x-barx )^2
6 16 - 19 = -3 9
7 17 - 19 = -2 4
8 18 - 19 = -1 1
9 20 - 19 = 1 1
10 21 - 19 = 2 4
Sum of the (x-barx)^2
9+4+1+1+4 = 19
Standard Deviation: Calculate the standard deviation
S = sqrt((sum(x- barx)^2 ) / (n-1))
S = sqrt( ( 9+4+1+1+4) / 4 )
S = sqrt (19 / 4 )
S = sqrt(4.75 )
S = 2.17944947
Practice Problems for Statistics Standard Deviation:
1. Find the statistics standard deviation for the following given data.37, 56, 54, 54, 26, 67, 12, 65 and 34.
Answer: Mean = 45
Standard Deviation = 18.714967272213
2. Calculate the statistics standard deviation for the following. 77, 56, 33, 87, 90, 23, 67, 80 and 99.
Answer: Mean = 68
Standard Deviation = 26.043233286211
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