Compute range and standard deviation for the following data : 71, 73, 74, 79, 81, 85, 92, 70, 75, 70.

1. Introduction

In this problem, we are given a set of data and tasked with computing the range and standard deviation. The range represents the difference between the highest and lowest values in the dataset, providing a measure of the spread or dispersion of the data. The standard deviation quantifies the average deviation of data points from the mean, providing a measure of the variability or dispersion of the data around the mean. Let's calculate these statistical measures for the given data.

2. Range Calculation

To calculate the range, we first need to determine the highest and lowest values in the dataset. Then, we find the difference between these two values.

Highest value = 92
Lowest value = 70

Range = Highest value – Lowest value
= 92 – 70
= 22

So, the range of the given data is 22.

3. Standard Deviation Calculation

To calculate the standard deviation, we follow these steps:

Step 1: Calculate the Mean

First, we need to calculate the mean of the dataset. The mean is the average value of the data.

Mean = (71 + 73 + 74 + 79 + 81 + 85 + 92 + 70 + 75 + 70) / 10
= 780 / 10
= 78

Step 2: Calculate Deviations

Next, we calculate the deviation of each data point from the mean. Deviation is the difference between each data point and the mean.

Deviation from mean:
71 – 78 = -7
73 – 78 = -5
74 – 78 = -4
79 – 78 = 1
81 – 78 = 3
85 – 78 = 7
92 – 78 = 14
70 – 78 = -8
75 – 78 = -3
70 – 78 = -8

Step 3: Square Deviations

Then, we square each deviation to eliminate negative values and emphasize differences from the mean.

Squared deviations:
(-7)^2 = 49
(-5)^2 = 25
(-4)^2 = 16
(1)^2 = 1
(3)^2 = 9
(7)^2 = 49
(14)^2 = 196
(-8)^2 = 64
(-3)^2 = 9
(-8)^2 = 64

Step 4: Calculate Variance

Next, we calculate the variance by finding the average of the squared deviations.

Variance = (49 + 25 + 16 + 1 + 9 + 49 + 196 + 64 + 9 + 64) / 10
= 492 / 10
= 49.2

Step 5: Calculate Standard Deviation

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

Standard Deviation = √(49.2)
≈ 7.01

So, the standard deviation of the given data is approximately 7.01.

4. Conclusion

In conclusion, we have calculated the range and standard deviation for the given dataset. The range is 22, indicating the difference between the highest and lowest values in the dataset. The standard deviation is approximately 7.01, providing a measure of the variability or dispersion of the data around the mean. These statistical measures offer valuable insights into the spread and variability of the data, aiding in data analysis and interpretation.