What is absolute dispersion ? Compute standard deviation for the following data : 2, 12, 14, 17, 18, 10, 9, 7, 1, 3.

1. Understanding Absolute Dispersion

Absolute dispersion measures the extent to which individual data points deviate from a central value, such as the mean or median, without regard to their direction. It provides information about the spread or variability of the data values within a dataset. Common measures of absolute dispersion include the range, mean absolute deviation (MAD), and standard deviation.

2. Computing Standard Deviation

Standard deviation is a widely used measure of absolute dispersion that quantifies the average distance of data points from the mean. It takes into account both the magnitude and direction of deviations from the mean, providing a comprehensive summary of the variability of data values within a dataset.

Step 1: Calculate the Mean

First, calculate the mean (average) of the given data set:

Mean = (2 + 12 + 14 + 17 + 18 + 10 + 9 + 7 + 1 + 3) / 10
Mean = 93 / 10
Mean = 9.3

Step 2: Calculate Deviations from the Mean

Next, calculate the deviation of each data point from the mean:

Deviation from Mean = Data Point – Mean

Deviation from Mean:
2 – 9.3 = -7.3
12 – 9.3 = 2.7
14 – 9.3 = 4.7
17 – 9.3 = 7.7
18 – 9.3 = 8.7
10 – 9.3 = 0.7
9 – 9.3 = -0.3
7 – 9.3 = -2.3
1 – 9.3 = -8.3
3 – 9.3 = -6.3

Step 3: Square the Deviations

Square each deviation to eliminate negative values and emphasize differences from the mean:

Squared Deviation = (Deviation from Mean)^2

Squared Deviation:
(-7.3)^2 = 53.29
(2.7)^2 = 7.29
(4.7)^2 = 22.09
(7.7)^2 = 59.29
(8.7)^2 = 75.69
(0.7)^2 = 0.49
(-0.3)^2 = 0.09
(-2.3)^2 = 5.29
(-8.3)^2 = 68.89
(-6.3)^2 = 39.69

Step 4: Calculate the Variance

Compute the variance by finding the average of the squared deviations:

Variance = Σ(Squared Deviation) / N
Variance = (53.29 + 7.29 + 22.09 + 59.29 + 75.69 + 0.49 + 0.09 + 5.29 + 68.89 + 39.69) / 10
Variance = 331.80 / 10
Variance = 33.18

Step 5: Calculate the Standard Deviation

Finally, calculate the standard deviation by taking the square root of the variance:

Standard Deviation = √Variance
Standard Deviation = √33.18
Standard Deviation ≈ 5.76

Conclusion

The standard deviation of the given data set {2, 12, 14, 17, 18, 10, 9, 7, 1, 3} is approximately 5.76. Standard deviation provides a measure of the average distance of data points from the mean, indicating the extent of variability or dispersion within the dataset.