Compute range and standard deviation for the following data : 81, 32, 61, 72, 74, 75, 76, 71, 84, 85.

1. Introduction

In this task, we will compute the range and standard deviation for the given dataset.

2. Range Calculation

The range is the difference between the highest and lowest values in a dataset. It provides a measure of the spread or variability of the data.

First, let's arrange the data in ascending order:

32, 61, 71, 72, 74, 75, 76, 81, 84, 85

The lowest value is 32, and the highest value is 85.

Range = Highest value – Lowest value

Range = 85 – 32

Range = 53

Therefore, the range of the given dataset is 53.

3. Standard Deviation Calculation

The standard deviation measures the dispersion or spread of data points around the mean. It indicates the average deviation of individual data points from the mean.

The formula for calculating the standard deviation (σ) is:

[ \sigma = \sqrt{\frac{\sum{(X – \bar{X})^2}}{N}} ]

Where:

• ( X ) = Each individual data point
• ( \bar{X} ) = Mean of the dataset
• ( N ) = Total number of data points

First, let's calculate the mean ( ( \bar{X} ) ) of the dataset:

[ \bar{X} = \frac{81 + 32 + 61 + 72 + 74 + 75 + 76 + 71 + 84 + 85}{10} ]

[ \bar{X} = \frac{701}{10} ]

[ \bar{X} = 70.1 ]

Now, let's calculate the sum of squares of deviations from the mean:

[ \sum{(X – \bar{X})^2} = (81 – 70.1)^2 + (32 – 70.1)^2 + (61 – 70.1)^2 + (72 – 70.1)^2 + (74 – 70.1)^2 + (75 – 70.1)^2 + (76 – 70.1)^2 + (71 – 70.1)^2 + (84 – 70.1)^2 + (85 – 70.1)^2 ]

[ \sum{(X – \bar{X})^2} = 113.61 + 1218.01 + 92.41 + 2.89 + 14.44 + 27.04 + 37.21 + 1.21 + 170.84 + 225.00 ]

[ \sum{(X – \bar{X})^2} = 1892.66 ]

Now, let's plug the values into the standard deviation formula:

[ \sigma = \sqrt{\frac{1892.66}{10}} ]

[ \sigma = \sqrt{189.266} ]

[ \sigma ≈ 13.75 ]

Therefore, the standard deviation of the given dataset is approximately 13.75.

4. Summary

• Range: 53
• Standard Deviation: Approximately 13.75

5. Conclusion

In conclusion, the range and standard deviation of the given dataset have been calculated. These measures provide insights into the variability and dispersion of data points around the mean. The range indicates the difference between the highest and lowest values, while the standard deviation measures the average deviation of individual data points from the mean. These measures are useful for understanding the spread and distribution of numerical data.