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Chapter 3: Problem 5

Find the population variance and standard deviation or the sample variance and standard deviation as indicated. $$ \text { Sample: } 20,13,4,8,10 $$

Short Answer

Expert verified
The sample variance is 36, and the sample standard deviation is 6.

Step by step solution

01

- Calculate the Sample Mean

First, find the mean \(\bar{x}\) of the sample. Add all the sample values and divide by the number of samples: \[ \bar{x} = \frac{20 + 13 + 4 + 8 + 10}{5} = \frac{55}{5} = 11 \]
02

- Calculate Each Deviation from the Mean

Compute the deviation of each sample value from the mean \(\bar{x}\): \[ 20 - 11 = 9 \]\[ 13 - 11 = 2 \]\[ 4 - 11 = -7 \]\[ 8 - 11 = -3 \]\[ 10 - 11 = -1 \]
03

- Square Each Deviation

Square each of the deviations calculated in Step 2: \[ (20 - 11)^2 = 81 \]\[ (13 - 11)^2 = 4 \]\[ (4 - 11)^2 = 49 \]\[ (8 - 11)^2 = 9 \]\[ (10 - 11)^2 = 1 \]
04

- Sum the Squared Deviations

Add up all the squared deviations: \[ 81 + 4 + 49 + 9 + 1 = 144 \]
05

- Calculate the Sample Variance

The sample variance \(s^2\) is found by dividing the sum of the squared deviations by the number of samples minus one (n-1): \[ s^2 = \frac{144}{5 - 1} = \frac{144}{4} = 36 \]
06

- Calculate the Sample Standard Deviation

The sample standard deviation \(s\) is the square root of the sample variance: \[ s = \sqrt{36} = 6 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean Calculation
To start any variance or standard deviation calculation, you must first find the sample mean. The sample mean \(\bar{x}\) is the average of your sample data points. Add all your sample values together and then divide by the total number of samples.

For instance, with the sample [20, 13, 4, 8, 10], the sum of the values is 55, and there are 5 data points. So, the sample mean \(\bar{x}\) is calculated as: \[ \bar{x} = \frac{20 + 13 + 4 + 8 + 10}{5} = \frac{55}{5} = 11 \]

This mean provides a reference point from which deviations are measured.
Deviation from the Mean
The next step is to determine how each sample value deviates from the mean. This is simply the difference between each data point and the sample mean \(\bar{x}\).

For each value in our sample, subtract the mean:
  • \(20 - 11 = 9\)
  • \(13 - 11 = 2\)
  • \(4 - 11 = -7\)
  • \(8 - 11 = -3\)
  • \(10 - 11 = -1\)

These deviations show how far each value is from the average.
Squared Deviations
Having the deviations is not enough; we need to neutralize the effect of negative differences. We achieve this by squaring each deviation.

For the deviations calculated above:
  • \((20 - 11)^2 = 81\)
  • \((13 - 11)^2 = 4\)
  • \((4 - 11)^2 = 49\)
  • \((8 - 11)^2 = 9\)
  • \((10 - 11)^2 = 1\)

Squaring removes negative signs and ensures all deviations are treated equally regardless of their direction.
Sum of Squared Deviations
To find out how spread out the values in your sample are, sum up all the squared deviations.

For our sample, add together the squared deviations: \[ 81 + 4 + 49 + 9 + 1 = 144 \]
This sum, called the sum of squared deviations, tells you the total variability within your data set.
Sample Variance Formula
Once you have the sum of squared deviations, you need to calculate the sample variance \(s^2\). The sample variance is found by dividing the sum of the squared deviations by the number of samples minus one (\(n - 1\)).

For our sample: \[ s^2 = \frac{144}{5 - 1} = \frac{144}{4} = 36 \]
The sample variance gives us an average of the squared deviations, revealing how much the values in the sample differ from the mean.
Sample Standard Deviation Calculation
The standard deviation \(s\) is simply the square root of the sample variance. It returns the measure of spread back to the same units as the original data.

For our sample: \[ s = \sqrt{36} = 6 \]
The sample standard deviation provides a concrete understanding of the variability among the data points. A smaller standard deviation means data points are close to the mean, while a larger one indicates they are spread out.

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Most popular questions from this chapter

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