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Chapter 3: Problem 8

Find the population variance and standard deviation or the sample variance and standard deviation as indicated. $$ \text { Population: } 1,19,25,15,12,16,28,13,6 $$

Short Answer

Expert verified
Variance: 64, Standard Deviation: 8

Step by step solution

01

- Determine the Mean

Add all the values together and divide by the number of values to find the mean (average) of the population.\[ \text{Mean} = \frac{1 + 19 + 25 + 15 + 12 + 16 + 28 + 13 + 6}{9} \]\[ \text{Mean} = \frac{135}{9} = 15 \]
02

- Subtract the Mean and Square the Result

Subtract the mean from each number and square the result.\( (1 - 15)^2 = (-14)^2 = 196 \)\( (19 - 15)^2 = 4^2 = 16 \)\( (25 - 15)^2 = 10^2 = 100 \)\( (15 - 15)^2 = 0^2 = 0 \)\( (12 - 15)^2 = (-3)^2 = 9 \)\( (16 - 15)^2 = 1^2 = 1 \)\( (28 - 15)^2 = 13^2 = 169 \)\( (13 - 15)^2 = (-2)^2 = 4 \)\( (6 - 15)^2 = (-9)^2 = 81 \)
03

- Sum the Squared Differences

Add all the squared differences together.\[ 196 + 16 + 100 + 0 + 9 + 1 + 169 + 4 + 81 = 576 \]
04

- Calculate the Variance

Divide the sum of the squared differences by the number of values in the population.\[ \text{Variance} = \frac{576}{9} = 64 \]
05

- Calculate the Standard Deviation

Take the square root of the variance to get the standard deviation.\[ \text{Standard Deviation} = \sqrt{64} = 8 \]

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Key Concepts

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

mean
The mean is the average of a set of numbers. To find it, you add up all the values and then divide by the number of values. For our population:
1, 19, 25, 15, 12, 16, 28, 13, 6
we add these up to get 135 and divide by 9 (since there are 9 numbers).
So, Mean = 15.
Finding the mean helps us understand the central tendency of the data, giving us an idea of where the 'middle' value is.
squared differences
Next, we need to measure how each value differs from the mean and then square that difference. Squaring the difference removes negative values and emphasizes larger differences.

For example, (1 - 15)^2 = (-14)^2 = 196
Take each number:
  • (19 - 15)^2 = 16
  • (25 - 15)^2 = 100
  • (15 - 15)^2 = 0
  • (12 - 15)^2 = 9
  • (16 - 15)^2 = 1
  • (28 - 15)^2 = 169
  • (13 - 15)^2 = 4
  • (6 - 15)^2 = 81
These squared differences tell us how spread out the data is around the mean.
variance
Variance gives us a measure of how much the numbers in the data set differ from the mean.
To find variance, we sum up all the squared differences we calculated and then divide by the number of values.
Sum of squared differences = 576

Then divide by 9 (the number of values), so:
Variance = 64
This step gives us a numerical measurement of the spread in the population. A higher variance means data points are more spread out from the mean, whereas a lower variance indicates they are closer.
standard deviation
Standard deviation is the square root of variance and provides a measure of the spread of data points around the mean in the same unit as the original data.
First, we calculated the variance to be 64.
Taking the square root of the variance (which is \( \sqrt{64} \)), we get:
Standard Deviation = 8

The standard deviation is particularly useful because it's in the same units as our original data values, making it easier to interpret and compare. It tells us, on average, how much each data point differs from the mean.

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Most popular questions from this chapter

Fast Pass In 2000, the Walt Disney Company created the "fast pass"-a ticket issued to a rider for a popular ride and the rider is told to return at a specific time during the day. At that time, the rider is allowed to bypass the regular line, thereby reducing the wait time for that particular rider. When compared to wait times prior to creating the fast pass, overall wait times for rides at the park increased, on average, yet patrons to the park indicated they were happier with the fast-pass system. Use the concepts of central tendency and dispersion to explain why.

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