Compute average deviation and standard deviation for the following data : 7, 4, 3, 10, 9

1. Calculation of Average Deviation

Average deviation measures the average distance of each data point from the mean of the dataset. It is computed by finding the absolute difference between each data point and the mean, summing these differences, and then dividing by the total number of data points.

Step 1: Calculate the Mean

Mean = (Sum of all values) / (Total number of values)
Mean = (7 + 4 + 3 + 10 + 9) / 5
Mean = 33 / 5
Mean = 6.6

Step 2: Calculate the Absolute Deviation for Each Data Point

Absolute deviation = |Data point – Mean|

For the given dataset:

• Absolute deviation from the mean for 7 = |7 – 6.6| = 0.4
• Absolute deviation from the mean for 4 = |4 – 6.6| = 2.6
• Absolute deviation from the mean for 3 = |3 – 6.6| = 3.6
• Absolute deviation from the mean for 10 = |10 – 6.6| = 3.4
• Absolute deviation from the mean for 9 = |9 – 6.6| = 2.4

Step 3: Calculate the Sum of Absolute Deviations

Sum of absolute deviations = 0.4 + 2.6 + 3.6 + 3.4 + 2.4 = 12.4

Step 4: Calculate the Average Deviation

Average deviation = (Sum of absolute deviations) / (Total number of data points)
Average deviation = 12.4 / 5
Average deviation = 2.48

Average Deviation = 2.48

2. Calculation of Standard Deviation

Standard deviation measures the spread or dispersion of data points around the mean of the dataset. It is computed by taking the square root of the average of the squared differences between each data point and the mean.

Step 1: Calculate the Squared Deviation for Each Data Point

Squared deviation = (Data point – Mean)^2

For the given dataset:

• Squared deviation for 7 = (7 – 6.6)^2 = 0.16
• Squared deviation for 4 = (4 – 6.6)^2 = 6.76
• Squared deviation for 3 = (3 – 6.6)^2 = 13.96
• Squared deviation for 10 = (10 – 6.6)^2 = 11.56
• Squared deviation for 9 = (9 – 6.6)^2 = 5.76

Step 2: Calculate the Sum of Squared Deviations

Sum of squared deviations = 0.16 + 6.76 + 13.96 + 11.56 + 5.76 = 38.2

Step 3: Calculate the Variance

Variance = (Sum of squared deviations) / (Total number of data points)
Variance = 38.2 / 5
Variance = 7.64

Step 4: Calculate the Standard Deviation

Standard deviation = √Variance
Standard deviation = √7.64
Standard deviation ≈ 2.76

Standard Deviation ≈ 2.76

Conclusion

In summary, for the given dataset:

• Average deviation = 2.48
• Standard deviation ≈ 2.76

These measures provide insights into the dispersion or variability of data points around the mean. The average deviation measures the average distance of data points from the mean, while the standard deviation quantifies the spread of data points in the dataset.