What is standard deviation ? Compute standard deviation for the following data : 16, 20, 17, 18, 19, 21

Understanding Standard Deviation

Standard deviation is a measure of the dispersion or spread of a set of data points around the mean. It indicates how much individual data points deviate from the mean of the data set. A higher standard deviation indicates greater variability, while a lower standard deviation indicates less variability.

1. Calculation of Mean

The first step in calculating the standard deviation is to compute the mean (average) of the data set. The mean is calculated by summing up all the values in the data set and dividing by the total number of values.

For the given data set: 16, 20, 17, 18, 19, 21

[ \text{Sum of values} = 16 + 20 + 17 + 18 + 19 + 21 = 111 ]

[ \text{Number of values} = 6 ]

[ \text{Mean} = \frac{111}{6} = 18.5 ]

So, the mean of the given data set is ( 18.5 ).

2. Calculation of Variance

The variance is calculated by taking the average of the squared differences between each data point and the mean.

[ \text{Variance} = \frac{\sum{(x_i – \bar{x})^2}}{N} ]

Where:

• ( x_i ) represents each individual data point
• ( \bar{x} ) represents the mean of the data set
• ( N ) represents the total number of data points

For the given data set:

[ \text{Variance} = \frac{(16-18.5)^2 + (20-18.5)^2 + (17-18.5)^2 + (18-18.5)^2 + (19-18.5)^2 + (21-18.5)^2}{6} ]

[ = \frac{(-2.5)^2 + (1.5)^2 + (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (2.5)^2}{6} ]

[ = \frac{6.25 + 2.25 + 2.25 + 0.25 + 0.25 + 6.25}{6} ]

[ = \frac{17.5}{6} ]

[ = 2.9167 ]

So, the variance of the given data set is approximately ( 2.9167 ).

3. Calculation of Standard Deviation

The standard deviation is the square root of the variance. It represents the average deviation of data points from the mean.

[ \text{Standard Deviation} = \sqrt{\text{Variance}} ]

[ = \sqrt{2.9167} ]

[ \approx 1.7078 ]

So, the standard deviation of the given data set is approximately ( 1.7078 ).

Conclusion

Standard deviation is a measure of the dispersion or spread of a set of data points around the mean. By calculating the mean, variance, and standard deviation, researchers can quantify the variability within a data set and gain insights into the distribution of values. In the case of the given data set, the standard deviation provides information about the average deviation of individual data points from the mean.