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Chapter 3: Problem 13

Find the population variance and standard deviation or the sample variance and standard deviation as indicated. Population: 3,6,10,12,14

Short Answer

Expert verified
Variance = 16, Standard Deviation = 4

Step by step solution

01

Calculate the Mean

First, find the mean (average) of the population. To do this, sum all the numbers and then divide by the total number of values. Mean, ewline \(\bar{x} = \frac{3 + 6 + 10 + 12 + 14}{5} = \frac{45}{5} = 9\)
02

Subtract the Mean and Square the Result

Subtract the mean from each number and then square the result. \((3 - 9)^2 = 36\) \((6 - 9)^2 = 9\) \((10 - 9)^2 = 1\) \((12 - 9)^2 = 9\) \((14 - 9)^2 = 25\)
03

Sum the Squared Results

Sum all the squared deviations. \(36 + 9 + 1 + 9 + 25 = 80\)
04

Divide by the Number of Values (Population Variance)

To find the population variance, divide the sum of the squared deviations by the number of values. \(\text{Variance, } \sigma^2 = \frac{80}{5} = 16\)
05

Calculate the Standard Deviation

The standard deviation is the square root of the variance. \(\text{Standard Deviation, } \sigma = \sqrt{16} = 4\)

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Key Concepts

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

mean calculation
To grasp population statistics, understanding how to calculate the mean is essential. The mean acts as a central point for a set of numbers and is the sum of those numbers divided by the total count. This average gives a general idea of the overall magnitude of the dataset.
In our example, we have the numbers 3, 6, 10, 12, and 14. To find the mean, sum these numbers: 3 + 6 + 10 + 12 + 14 = 45. Then, divide by 5 (the number of values) to get \(\bar{x} = 9\). This mean represents a balance point where the values above and below it even out.
variance calculation
Variance measures the spread of values in a dataset around the mean. It tells you how much the numbers differ from the mean. Here鈥檚 a step-by-step process to calculate it:
1. Subtract the mean from each number:
\((3 - 9)^2 = 36\)
\((6 - 9)^2 = 9\)
\((10 - 9)^2 = 1\)
\((12 - 9)^2 = 9\)
\((14 - 9)^2 = 25\)
2. Square each result.
3. Sum all the squared results. In our example, \(36+ 9 + 1 + 9 + 25 = 80\)
4. Divide this sum by the total number of values (for population variance), so \( \frac{80}{5} = 16 \)
This process shows how the values are dispersed around the mean, with the calculated variance being 16.
standard deviation calculation
Standard deviation simplifies the concept of variance into a more interpretable value. It鈥檚 the square root of the variance, which translates the units back to the original dataset.
In our example, with a variance of 16, the standard deviation is calculated as: \(\text{Standard Deviation, } \sigma = \sqrt{16} = 4\).
This value shows that, on average, each number in the population deviates from the mean (9) by about 4 units. The smaller the standard deviation, the closer the numbers are to the mean.
population statistics
Population statistics involves analyzing every member within a dataset or population. This differs from sample statistics, which only examines a subset.
The dataset in this exercise (3, 6, 10, 12, 14) represents a full population, and we aim to find the population variance and standard deviation. These results inform us about the data's overall characteristics.
By calculating the mean, variance, and standard deviation, we gain insights into the central tendency and dispersion of the entire dataset. This statistical analysis is helpful in many fields, from scientific research to business analytics, as it provides a precise understanding of the population's behavior.

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Most popular questions from this chapter

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