For the following data, determine the range and standard deviation:
9, 12, 13, 17, 8, 16, 10, 4, 7, 11, and 12.
Compute range and standard deviation for the following data : 8, 16, 10, 4, 7, 11, 13, 17, 9, 12.
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For the following data, determine the range and standard deviation:
9, 12, 13, 17, 8, 16, 10, 4, 7, 11, and 12.
Range and Standard Deviation Calculation
1. Range:
The range is the difference between the highest and lowest values in the data set. It provides a measure of the spread or variability of the data.
To calculate the range:
Range = 13
2. Standard Deviation:
Standard deviation is a measure of the dispersion or spread of a set of values. It quantifies the average deviation of individual data points from the mean.
To calculate the standard deviation, follow these steps:
a. Calculate the Mean:
First, find the mean of the data set by adding up all the values and dividing by the total number of values.
Mean = (8 + 16 + 10 + 4 + 7 + 11 + 13 + 17 + 9 + 12) / 10
= 107 / 10
= 10.7
b. Calculate the Deviation from the Mean:
Next, find the deviation of each data point from the mean by subtracting the mean from each value.
Deviation from Mean = (8 – 10.7), (16 – 10.7), (10 – 10.7), (4 – 10.7), (7 – 10.7), (11 – 10.7), (13 – 10.7), (17 – 10.7), (9 – 10.7), (12 – 10.7)
= -2.7, 5.3, -0.7, -6.7, -3.7, 0.3, 2.3, 6.3, -1.7, 1.3
c. Square the Deviations:
Square each deviation to eliminate negative values and emphasize differences from the mean.
Squared Deviation = (-2.7)^2, (5.3)^2, (-0.7)^2, (-6.7)^2, (-3.7)^2, (0.3)^2, (2.3)^2, (6.3)^2, (-1.7)^2, (1.3)^2
= 7.29, 28.09, 0.49, 44.89, 13.69, 0.09, 5.29, 39.69, 2.89, 1.69
d. Calculate the Variance:
Find the average of the squared deviations, known as the variance.
Variance = (7.29 + 28.09 + 0.49 + 44.89 + 13.69 + 0.09 + 5.29 + 39.69 + 2.89 + 1.69) / 10
= 143.01 / 10
= 14.301
e. Calculate the Standard Deviation:
Take the square root of the variance to find the standard deviation.
Standard Deviation = √14.301
≈ 3.78
Standard Deviation ≈ 3.78
Conclusion:
In summary, for the given data set:
The range provides a measure of the spread of the data, while the standard deviation quantifies the average deviation of individual data points from the mean. Both measures help to assess the variability or dispersion of the data set.