For the following data, determine the range and standard deviation: 2, 10, 7, 6, 5, 14, 12, 7, 8, 1.
Compute range and standard deviation for the following data : 2, 10, 7, 6, 5, 14, 12, 7, 8, 1.
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For the following data, determine the range and standard deviation: 2, 10, 7, 6, 5, 14, 12, 7, 8, 1.
1. Range
The range of a dataset is the difference between the highest and lowest values in the dataset. It provides a measure of the spread or variability of the data.
To compute the range for the given dataset:
Arrange the dataset in ascending order:
1, 2, 5, 6, 7, 7, 8, 10, 12, 14
Calculate the difference between the highest and lowest values:
Range = Maximum value – Minimum value
Range = 14 – 1
Range = 13
Therefore, the range of the given dataset is 13.
2. Standard Deviation
The standard deviation measures the average distance of each data point from the mean of the dataset. It provides a measure of the dispersion or variability of the data points around the mean.
To compute the standard deviation for the given dataset:
Calculate the mean of the dataset:
Mean = (2 + 10 + 7 + 6 + 5 + 14 + 12 + 7 + 8 + 1) / 10
Mean = 72 / 10
Mean = 7.2
Calculate the squared differences between each data point and the mean:
(2 – 7.2)^2 = 29.16
(10 – 7.2)^2 = 7.84
(7 – 7.2)^2 = 0.04
(6 – 7.2)^2 = 14.44
(5 – 7.2)^2 = 4.84
(14 – 7.2)^2 = 46.24
(12 – 7.2)^2 = 23.04
(7 – 7.2)^2 = 0.04
(8 – 7.2)^2 = 0.64
(1 – 7.2)^2 = 38.44
Calculate the sum of the squared differences:
29.16 + 7.84 + 0.04 + 14.44 + 4.84 + 46.24 + 23.04 + 0.04 + 0.64 + 38.44 = 164.36
Divide the sum of squared differences by the number of data points (N) to get the variance:
Variance = Sum of squared differences / N
Variance = 164.36 / 10
Variance = 16.436
Take the square root of the variance to get the standard deviation:
Standard Deviation = √Variance
Standard Deviation = √16.436
Standard Deviation ≈ 4.06
Therefore, the standard deviation of the given dataset is approximately 4.06.
Conclusion
In summary, for the given dataset:
These measures provide insights into the spread and variability of the data, helping to understand the distribution of values and assess the consistency or dispersion of the dataset.