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Magazine Chapter 3: Problem 9
Find the population variance and standard deviation or the sample variance and standard deviation as indicated. $$ \text { Sample: } 6,52,13,49,35,25,31,29,31,29 $$
Short Answer
Expert verified
Sample variance is 196, and sample standard deviation is 14.
Step by step solution
01
Understand the Problem
Determine whether to find the sample variance and standard deviation. Given a sample: 6, 52, 13, 49, 35, 25, 31, 29, 31, 29.
02
Calculate the Sample Mean
Find the mean of the sample data. Formula: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]Substitute the given values: \[ \bar{x} = \frac{6 + 52 + 13 + 49 + 35 + 25 + 31 + 29 + 31 + 29}{10} = \frac{300}{10} = 30 \]
03
Calculate the Squared Differences
For each sample data point, subtract the mean and square the result: \((6 - 30)^2 = 576\) \((52 - 30)^2 = 484\)\((13 - 30)^2 = 289\) \((49 - 30)^2 = 361\)\((35 - 30)^2 = 25\)\((25 - 30)^2 = 25\)\((31 - 30)^2 = 1\)\((29 - 30)^2 = 1\)\((31 - 30)^2 = 1\)\((29 - 30)^2 = 1\)
04
Sum of Squared Differences
Add all the squared differences together:\[ 576 + 484 + 289 + 361 + 25 + 25 + 1 + 1 + 1 + 1 = 1764 \]
05
Calculate the Sample Variance
Use the formula for sample variance (denoted as \( s^2 \)):\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]Substitute the values:\[ s^2 = \frac{1764}{10-1} = \frac{1764}{9} = 196 \]
06
Calculate the Sample Standard Deviation
Use the formula for sample standard deviation (denoted as \( s \)):\[ s = \sqrt{s^2} \]Substitute the found variance value:\[ s = \sqrt{196} = 14 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sample Mean
To start with the calculation of variance and standard deviation, finding the **sample mean** is crucial. The sample mean, also referred to as \( \bar{x} \), is the average value of the sample data points. The formula to find the sample mean is simple: sum up all the sample data points and divide by the number of data points (sample size). For instance, given the sample 6, 52, 13, 49, 35, 25, 31, 29, 31, 29, the mean would be calculated as follows: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \], where \( n \) is the number of data points. Substituting the values, we get \[ \bar{x} = \frac{6 + 52 + 13 + 49 + 35 + 25 + 31 + 29 + 31 + 29}{10} = 30 \].
Squared Differences
Next, you must calculate the **squared differences** for each data point in your sample. This step is essential because it helps measure how spread out the data points are around the mean. For each data point, you subtract the sample mean and square the result. For example: for the first data point (6), you calculate \( (6-30)^2 = 576 \). Repeating this for all data points, we get a set of squared differences: \[ (52-30)^2 = 484, \, (13-30)^2 = 289, \, (49-30)^2 = 361, \, (35-30)^2 = 25, \, (25-30)^2 = 25, \, (31-30)^2 = 1, \, (29-30)^2 = 1, \, (31-30)^2 = 1, \, (29-30)^2 = 1 \].
Sum of Squared Differences
After calculating the squared differences, the next step is to find the **sum of squared differences**. This step consolidates all the squared deviations to provide an overview of the total dispersion in the dataset. Simply add all the squared differences together: \[ 576 + 484 + 289 + 361 + 25 + 25 + 1 + 1 + 1 + 1 = 1764 \]. This total represents the cumulative spread of the data points around the mean.
Variance Calculation
Once you have the sum of squared differences, you can calculate the **sample variance**. Variance gives a measure of how data points differ from the mean. The formula for sample variance is: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]. We use \( n-1 \) instead of \( n \) to account for the degrees of freedom in the sample. For our example, we substitute the values: \[ s^2 = \frac{1764}{10-1} = \frac{1764}{9} = 196 \]. So, the sample variance is 196.
Standard Deviation Formula
Finally, you find the **sample standard deviation**, which is the square root of the sample variance. Standard deviation is a key statistic because it gives us a sense of the average distance of data points from the mean. The formula is: \[ s = \sqrt{s^2} \]. For our dataset, substituting the variance value, we get: \[ s = \sqrt{196} = 14 \]. This means the average distance of the data points from the mean is 14.
