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Chapter 3: Problem 14

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

Short Answer

Expert verified
Variance = 64, Standard Deviation = 8

Step by step solution

01

- Calculate the Mean

Sum all the population values and divide by the number of values.Mean \(\bar{x} = \frac{1 + 19 + 25 + 15 + 12 + 16 + 28 + 13 + 6}{9} = \frac{135}{9} = 15\)So, the mean is 15.
02

- Find Each Deviation

Subtract the mean from each value to get the deviation.Deviations: (1-15), (19-15), (25-15), (15-15), (12-15), (16-15), (28-15), (13-15), (6-15)This results in: -14, 4, 10, 0, -3, 1, 13, -2, -9.
03

- Square Each Deviation

Square each of the deviation values.Squares: \(((-14)^2), (4^2), (10^2), (0^2), ((-3)^2), (1^2), (13^2), ((-2)^2), ((-9)^2)\)This results in: 196, 16, 100, 0, 9, 1, 169, 4, 81.
04

- Sum of Squared Deviations

Sum all the squared deviations.Sum = 196 + 16 + 100 + 0 + 9 + 1 + 169 + 4 + 81 = 576.
05

- Calculate the Variance

Divide the sum of squared deviations by the number of values (since it's the entire population).Variance \(\text{Var} = \frac{576}{9} = 64\)So, the variance is 64.
06

- Calculate the Standard Deviation

Take the square root of the variance to get the standard deviation.Standard Deviation \(\text{SD} = \sqrt{64} = 8\).So, the standard deviation is 8.

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Key Concepts

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

Mean Calculation
The first step in finding the population variance and standard deviation is to calculate the mean. The mean, represented often as \(\bar{x}\), is the average of all the values in a data set. To calculate the mean, you sum up all the values and then divide by the number of values in the population.

For our data set: 1, 19, 25, 15, 12, 16, 28, 13, 6

Here's how the calculation is done:

  • Add all the numbers: 1 + 19 + 25 + 15 + 12 + 16 + 28 + 13 + 6 = 135
  • Count the number of values: 9
  • Divide the sum by the number of values: \(\bar{x} = \frac{135}{9} = 15\)

So, the mean is 15.

This mean value is essential as it serves as a central reference point from which we measure the deviations in the next step.
Deviation Calculation
Once you have the mean, the next step is to find the deviations. A deviation is simply the difference between each value in the data set and the mean.

For example, if your value is 1 and the mean is 15, the deviation is 1 - 15 = -14.
  • Calculate for each value: (1-15), (19-15), (25-15), (15-15), (12-15), (16-15), (28-15), (13-15), (6-15)
  • This results in: -14, 4, 10, 0, -3, 1, 13, -2, -9

This list of deviations tells you how much each value differs from the mean.
Squared Deviations
To proceed to the next step, we need to square each deviation. Squaring eliminates negative signs and gives more weight to larger differences.

For example, squaring the deviation -14 yields:
  • \((-14)^2 = 196\)

  • Do this for all deviations: \((-14)^2, (4)^2, (10)^2, (0)^2, (-3)^2, (1)^2, (13)^2, (-2)^2, (-9)^2\)
  • This results in: 196, 16, 100, 0, 9, 1, 169, 4, 81

These squared deviations will then be used to calculate the sum of squared deviations.
Sum of Squared Deviations
Adding up all the squared deviations gives us the sum of squared deviations, which is crucial for further calculations.
  • Add all squared deviations: 196 + 16 + 100 + 0 + 9 + 1 + 169 + 4 + 81 = 576

This sum encapsulates how much variation exists within your data set, making it a key number for calculating both the variance and standard deviation.
Variance Calculation
Variance gauges the spread between numbers in a data set. To find the population variance, you divide the sum of squared deviations by the total number of values.
  • Variance \(\text{Var} = \frac{576}{9} = 64\)

So, the population variance is 64.

This variance value helps to understand how each data point's deviation relates to the overall data set.
Standard Deviation Calculation
The final step is calculating the standard deviation, which quantifies the amount of variation or dispersion in a data set. The standard deviation is the square root of the variance.
  • Standard Deviation \(\text{SD} = \sqrt{64} = 8\)

So, the standard deviation is 8.

This number provides a sense of how spread out the values in your data set are around the mean.

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Most popular questions from this chapter

Morningstar is a mutual fund rating agency. It ranks a fund's performance by using one to five stars. A one-star mutual fund is in the bottom \(20 \%\) of its investment class; a five-star mutual fund is in the top \(20 \%\) of its investment class Interpret the meaning of a four-star mutual fund.

True or False: When comparing two populations, the larger the standard deviation, the more dispersion the distribution has, provided that the variable of interest from the two populations has the same unit of measure.

Concepts and Vocabulary What does it mean if a statistic is resistant? Why is the median resistant, but the mean is not? Is the mode a resistant measure of center?

In Marissa's calculus course, attendance counts for \(5 \%\) of the grade, quizzes count for \(10 \%\) of the grade, exams count for \(60 \%\) of the grade, and the final exam counts for \(25 \%\) of the grade. Marissa had a \(100 \%\) average for attendance, \(93 \%\) for quizzes, \(86 \%\) for exams, and \(85 \%\) on the final. Determine Marissa's course average.

For each of the following situations, determine which measure of central tendency is most appropriate and justify your reasoning. (a) Average price of a home sold in Pittsburgh, Pennsylvania, in 2002 (b) Most popular major for students enrolled in a statistics course (c) Average test score when the scores are distributed symmetrically (d) Average test score when the scores are skewed right (e) Average income of a player in the National Football League (f) Most requested song at a radio station
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