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Magazine Chapter 3: Problem 6
Find the population variance and standard deviation or the sample variance and standard deviation as indicated. $$ \text { Sample: } 83,65,91,87,84 $$
Short Answer
Expert verified
Sample variance is 100. Sample standard deviation is 10.
Step by step solution
01
- Compute the Sample Mean
First, calculate the mean of the sample. The formula for the mean \( \bar{x} \) of a sample is: \[ \bar{x} = \frac{\text{sum of all observations}}{\text{number of observations}} \] \ \[\bar{x} = \frac{83 + 65 + 91 + 87 + 84}{5} = \frac{410}{5} = 82 \]
02
- Calculate Each Deviation from the Mean
Subtract the mean from each observation to find the deviation: \[ 83 - 82 = 1, \ 65 - 82 = -17, \ 91 - 82 = 9, \ 87 - 82 = 5, \ 84 - 82 = 2 \]
03
- Square Each Deviation
Square each of the deviations to eliminate negative values: \[ 1^2 = 1, \ (-17)^2 = 289, \ 9^2 = 81, \ 5^2 = 25, \ 2^2 = 4 \]
04
- Compute the Variance
Sum the squared deviations and divide by the number of observations minus one (since this is a sample): \[ S^2 = \frac{1 + 289 + 81 + 25 + 4}{5-1} = \frac{400}{4} = 100 \] The sample variance is 100.
05
- Compute the Standard Deviation
Finally, take the square root of the variance to get the standard deviation: \[ S = \sqrt{100} = 10 \] The sample standard deviation is 10.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sample Mean
To begin with, the sample mean, often called the average, gives us a central value for the data points in our sample. It鈥檚 calculated as the sum of all the observations divided by the number of observations.
The formula for the sample mean (\( \bar{x} \)) is:
\[ \bar{x} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
For example, if your sample data consists of the numbers 83, 65, 91, 87, and 84,
The formula for the sample mean (\( \bar{x} \)) is:
\[ \bar{x} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
For example, if your sample data consists of the numbers 83, 65, 91, 87, and 84,
- First, add up all these values: 83 + 65 + 91 + 87 + 84 = 410.
- Next, divide the total by the number of observations, which is 5: \[ \bar{x} = \frac{410}{5} = 82 \]
Deviation from the Mean
Deviation from the mean helps us understand how each individual observation in the sample differs from the central value (the mean).
To find the deviation, subtract the sample mean from each observation:
To find the deviation, subtract the sample mean from each observation:
- For 83: Deviation = 83 - 82 = 1
- For 65: Deviation = 65 - 82 = -17
- For 91: Deviation = 91 - 82 = 9
- For 87: Deviation = 87 - 82 = 5
- For 84: Deviation = 84 - 82 = 2
Squared Deviations
To find the squared deviations, simply square each of the deviations calculated in the previous step. This squaring is essential to eliminate negative values and emphasize larger deviations.
Here鈥檚 how it鈥檚 done:
Here鈥檚 how it鈥檚 done:
- For a deviation of 1: \[ 1^2 = 1 \]
- For a deviation of -17: \[ (-17)^2 = 289 \]
- For a deviation of 9: \[ 9^2 = 81 \]
- For a deviation of 5: \[ 5^2 = 25 \]
- For a deviation of 2: \[ 2^2 = 4 \]
Variance Calculation
Once we have the squared deviations, we can move on to calculating the variance. Variance provides a measure of how spread out the values in a sample are around the mean.
\[ S^2 = \frac{\sum (\text{Squared Deviations})}{\text{Number of Observations} - 1} \]
For our example:
\[ S^2 = \frac{\sum (\text{Squared Deviations})}{\text{Number of Observations} - 1} \]
For our example:
- Sum of squared deviations = 1 + 289 + 81 + 25 + 4 = 400
- Number of observations = 5
- Thus, \[ S^2 = \frac{400}{5 - 1} = \frac{400}{4} = 100 \]
Standard Deviation Calculation
Finally, the standard deviation gives us a measure of the spread of data points in their original units. It鈥檚 the square root of the variance.
\[ S = \sqrt{S^2} \]
For our sample:
\[ S = \sqrt{100} = 10 \]
This result indicates that, on average, the observations differ from the mean by about 10 units. The standard deviation helps us understand the degree of variation in our sample data.
Remember, both variance and standard deviation are key concepts in statistics, providing insights into the variability within datasets and aiding in further statistical analyses.
\[ S = \sqrt{S^2} \]
For our sample:
\[ S = \sqrt{100} = 10 \]
This result indicates that, on average, the observations differ from the mean by about 10 units. The standard deviation helps us understand the degree of variation in our sample data.
Remember, both variance and standard deviation are key concepts in statistics, providing insights into the variability within datasets and aiding in further statistical analyses.
