Solved Examples on Range and Standard Deviation
Example 1: Calculate the Range and standard deviation for the following dataset: 10,15,20,25,30.
Solution:
Range = (maximum value- minimum value)
( 30-10) = 20.
For Standard deviation following steps are used
Calculate Mean
We need to calculate the mean of the dataset before finding standard deviation,
Mean = (10+15+20+25+30)/5 = 20
Calculate the Deviations from the Mean
Deviation from the mean for each value = Value – Mean
Deviations: (-10), (-5), 0, 5, 10
Calculate the Squared Deviations
Squared deviation for each value = (Deviation from the mean)²
Squared deviations: 100, 25, 0, 25, 100
Calculate the Variance
Variance = (Sum of squared deviations) / (Number of values) = (100 + 25 + 0 + 25 + 100) / 5 = 250 / 5 = 50
Calculate the Standard Deviation
Standard deviation = √variance = √50
=7.07
Example 2: Consider the following set of numbers representing the daily temperatures (in degrees Celsius) for a week: 20, 22, 24, 23, 25, 21, 19.
Solution:
Arrange the numbers in ascending order: 19, 20, 21, 22, 23, 24, 25.
Range = Largest value – Smallest value = 25 – 19 = 6.
So, the range of the daily temperatures for the week is 6 degrees Celsius.
For Standard deviation following steps are used
Calculate Mean
Mean = (19 + 20 + 21 + 22 + 23 + 24 + 25) / 7 = 154 / 7 ≈ 22.
Calculate the Deviations from the Mean
Deviation from the mean for each temperature = Temperature – Mean
Deviations: -2, 0, 2, 1, 3, -1, -3
Calculate the Squared Deviations
Squared deviation for each temperature = (Deviation from the mean)²
Squared deviations: 4, 0, 4, 1, 9, 1, 9
Calculate the Variance
Variance = (Sum of squared deviations) / (Number of temperatures) = (4 + 0 + 4 + 1 + 9 + 1 + 9) / 7 = 28 / 7 = 4
Calculate the Standard Deviation
Standard deviation = √(Variance) = √4 = 2
So, the standard deviation of the daily temperatures for the week is approximately 2 degrees Celsius.
Differences between Range and Standard Deviation
In statistics, Range and standard deviation provide insight into the spread or dispersion of data points within the data set. Range and standard deviation are both measures of variability in a dataset, but they differ in their calculation and interpretation.
The purpose of this article is to know the difference between range and standard deviation for the students offering clarity on calculations.