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Chapter 3: Problem 4

Identify the symbols used for each of the following: (a) sample standard deviation; (b) population standard deviation; (c) sample variance; (d) population variance.

Short Answer

Expert verified
s, 蟽, s虏, 蟽虏

Step by step solution

01

- Sample Standard Deviation

The symbol for the sample standard deviation is typically denoted by the letter 's'.
02

- Population Standard Deviation

The symbol for the population standard deviation is represented by the Greek letter '蟽' (sigma).
03

- Sample Variance

The symbol for the sample variance is denoted by 's虏'.
04

- Population Variance

The symbol for the population variance is represented by '蟽虏' (sigma squared).

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Key Concepts

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

Sample Standard Deviation
Sample standard deviation helps us understand how spread out the data points in a sample are. We denote the sample standard deviation using the letter 's'.
It measures the extent to which individual data points deviate from the sample mean. This can be calculated with the formula:

\(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \)

Here, \(x_i\) represents each data point, \(\bar{x}\) is the sample mean, and \(n\) is the number of data points in the sample. Use sample standard deviation when you have a dataset that represents only a part of a larger population.
Population Standard Deviation
Population standard deviation tells us how spread out the data points are in an entire population. We represent the population standard deviation with the Greek letter '蟽'.
It calculates how much the data points differ from the population mean. You can find it with this formula:

\( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \)

In this formula, \(x_i\) is each data point, \(\mu\) is the population mean, and \(N\) is the total number of data points in the population. Use population standard deviation when your dataset includes every member of the population.
Sample Variance
Sample variance is a way to find out how much the data points in a sample spread out around the mean. We denote the sample variance as 's虏'.
It's essentially the average of the squared differences between each data point and the sample mean. The formula for sample variance is:

\( s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \)

This means that you first find the difference between each data point (\(x_i\)) and the sample mean (\(\bar{x}\)), square those differences, sum them up, and then divide by \(n - 1\). It's commonly used when you have a smaller subset of data.
Population Variance
Population variance helps us understand how data points in an entire population are spread out around the mean. We denote it using '蟽虏' (sigma squared).
This metric gives us the average of the squared differences between each data point and the population mean. The formula for population variance is:

\( \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \)

In this case, you calculate the variance by finding the difference between each data point (\(x_i\)) and the population mean (\(\mu\)), squaring these differences, summing them up, and then dividing by \(N\). Use population variance when your data includes every individual in the population.

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Most popular questions from this chapter

Refer to the frequency distribution in the given exercise and find the standard deviation by using the formula below, where \(x\) represents the class midpoint, \(f\) represents the class frequency, and \(n\) represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3-4 with the original list of data values: (Exercise 37) 11.5 years; (Exercise 38) 8.9 years; (Exercise 39) 59.5; (Exercise 40) 65.4. $$s=\sqrt{\frac{n\left[\Sigma\left(f \cdot x^{2}\right)\right]-[\Sigma(f \cdot x)]^{2}}{n(n-1)}}$$ $$ \begin{array}{|c|c|} \hline \text { Age (yr) of Best Actress } & \\ \text { When Oscar Was Won } & \text { Frequency } \\ \hline 20-29 & 29 \\ \hline 30-39 & 34 \\ \hline 40-49 & 14 \\ \hline 50-59 & 3 \\ \hline 60-69 & 5 \\ \hline 70-79 & 1 \\ \hline 80-89 & 1 \\ \hline \end{array} $$

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, \((b)\) median, (c) mode, (d) midrange, and then answer the given question. Listed below are selling prices (dollars) of TVs that are 60 inches or larger and rated as a "best buy" by Consumer Reports magazine. Are the resulting statistics representative of the population of all TVs that are 60 inches and larger? If you decide to buy one of these TVs, what statistic is most relevant, other than the measures of central tendency? \(\begin{array}{llllllllllll}1800 & 1500 & 1200 & 1500 & 1400 & 1600 & 1500 & 950 & 1600 & 1150 & 1500 & 1750\end{array}\)

Here are four of the Verizon data speeds (Mbps) from Figure 3-1: \(13.5,10.2,21.1,15.1\). Find the mean and median of these four values. Then find the mean and median after including a fifth value of 142 , which is an outlier. (One of the Verizon data speeds is \(14.2 \mathrm{Mbps}\), but 142 is used here as an error resulting from an entry with a missing decimal point.) Compare the two sets of results. How much was the mean affected by the inclusion of the outlier? How much is the median affected by the inclusion of the outlier?

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions. In the California Health Interview Survey, randomly selected adults are interviewed. One of the questions asks how many cigarettes are smoked per day, and results are listed below for 50 randomly selected respondents. How well do the results reflect the smoking behavior of California adults? \(\begin{array}{rrrrrrrrrrrrrrr}9 & 10 & 10 & 20 & 40 & 50 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & & & & & & & & & & \end{array}\)

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions. Listed below are the numbers of Atlantic hurricanes that occurred in each year. The data are listed in order by year, starting with the year 2000 . What important feature of the data is not revealed by any of the measures of variation? $$ \begin{array}{llllllllllllll} 8 & 9 & 8 & 7 & 9 & 15 & 5 & 6 & 8 & 4 & 12 & 7 & 8 & 2 \end{array} $$
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