Compute range and standard deviation for the following data : 70, 81, 89, 91, 98, 61, 25, 35, 40, 60.

1. Introduction

In statistical analysis, measures of dispersion are crucial in understanding the spread or variability within a dataset. Two common measures used for this purpose are the range and standard deviation. The range provides a simple indication of how spread out the values in a dataset are, while the standard deviation offers a more precise measure of dispersion, taking into account the variability of each data point from the mean.

2. Computing the Range

The range of a dataset is the difference between the maximum and minimum values. In this dataset:

Data: 70, 81, 89, 91, 98, 61, 25, 35, 40, 60

Maximum value = 98

Minimum value = 25

Range = Maximum value – Minimum value

Range = 98 – 25 = 73

Thus, the range of the given dataset is 73.

3. Computing the Mean

Before calculating the standard deviation, it's essential to compute the mean (average) of the dataset. The mean is the sum of all values divided by the total number of values.

Mean = (Sum of all values) / (Number of values)

Mean = (70 + 81 + 89 + 91 + 98 + 61 + 25 + 35 + 40 + 60) / 10

Mean = 670 / 10

Mean = 67

The mean of the given dataset is 67.

4. Computing the Deviations

Next, compute the deviations of each data point from the mean. The deviation of a data point is the difference between the data point and the mean.

Deviation = Data point – Mean

For the given dataset, the deviations are:

70 – 67 = 3

81 – 67 = 14

89 – 67 = 22

91 – 67 = 24

98 – 67 = 31

61 – 67 = -6

25 – 67 = -42

35 – 67 = -32

40 – 67 = -27

60 – 67 = -7

5. Squaring the Deviations

To compute the standard deviation, we square each deviation. Squaring the deviations ensures that negative deviations do not cancel out positive deviations when computing the average variability.

Squared Deviation = Deviation^2

For the given dataset, the squared deviations are:

3^2 = 9

14^2 = 196

22^2 = 484

24^2 = 576

31^2 = 961

(-6)^2 = 36

(-42)^2 = 1764

(-32)^2 = 1024

(-27)^2 = 729

(-7)^2 = 49

6. Computing the Variance

The variance is the average of the squared deviations. It gives a measure of the average variability of the dataset from the mean.

Variance = (Sum of squared deviations) / (Number of values)

Variance = (9 + 196 + 484 + 576 + 961 + 36 + 1764 + 1024 + 729 + 49) / 10

Variance = 5838 / 10

Variance = 583.8

7. Computing the Standard Deviation

Finally, the standard deviation is the square root of the variance. It represents the typical distance between each data point and the mean.

Standard Deviation = √(Variance)

Standard Deviation ≈ √(583.8)

Standard Deviation ≈ 24.15

Thus, the standard deviation of the given dataset is approximately 24.15.

Conclusion

In conclusion, the range of the dataset is 73, indicating the spread between the maximum and minimum values. The standard deviation, which provides a more precise measure of dispersion, is approximately 24.15, indicating the average variability of the dataset from the mean. These measures are essential in understanding the distribution and variability of data points in a dataset, providing valuable insights for analysis and decision-making.