# 2.7 Measures of the spread of the data -- rrc math 1020  (Page 7/25)

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## For any data set, no matter what the distribution of the data is:

• At least 75% of the data is within two standard deviations of the mean.
• At least 89% of the data is within three standard deviations of the mean.
• At least 95% of the data is within 4.5 standard deviations of the mean.
• This is known as Chebyshev's Rule.

## For data having a distribution that is bell-shaped and symmetric:

• Approximately 68% of the data is within one standard deviation of the mean.
• Approximately 95% of the data is within two standard deviations of the mean.
• More than 99% of the data is within three standard deviations of the mean.
• This is known as the Empirical Rule.
• It is important to note that this rule only applies when the shape of the distribution of the data is bell-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters.

## References

Data from Microsoft Bookshelf.

King, Bill.“Graphically Speaking.” Institutional Research, Lake Tahoe Community College. Available online at http://www.ltcc.edu/web/about/institutional-research (accessed April 3, 2013).

## Chapter review

The standard deviation can help you calculate the spread of data. There are different equations to use if are calculating the standard deviation of a sample or of a population.

• The Standard Deviation allows us to compare individual data or classes to the data set mean numerically.
• s = $\sqrt{\frac{{\sum }^{\text{​}}{\left(x-\overline{x}\right)}^{2}}{n-1}}$ or s = $\sqrt{\frac{{\sum }^{\text{​}}f{\left(x-\overline{x}\right)}^{2}}{n-1}}$ is the formula for calculating the standard deviation of a sample. To calculate the standard deviation of a population, we would use the population mean, μ , and the formula σ = $\sqrt{\frac{{\sum }^{\text{​}}{\left(x-\mu \right)}^{2}}{N}}$ or σ = $\sqrt{\frac{{\sum }^{\text{​}}f{\left(x-\mu \right)}^{2}}{N}}$ .

## Formula review

${s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}$ where

## Practice

Use the following information to answer the next two exercises : The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.
29; 37; 38; 40; 58; 67; 68; 69; 76; 86; 87; 95; 96; 96; 99; 106; 112; 127; 145; 150

Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth.

s = 34.5

Find the value that is one standard deviation below the mean.

Two baseball players, Fredo and Karl, on different teams wanted to find out who had the higher batting average when compared to his team. Which baseball player had the higher batting average when compared to his team?

Baseball Player Batting Average Team Batting Average Team Standard Deviation
Fredo 0.158 0.166 0.012
Karl 0.177 0.189 0.015

For Fredo: z = = –0.67

For Karl: z = = –0.8

Fredo’s z -score of –0.67 is higher than Karl’s z -score of –0.8. For batting average, higher values are better, so Fredo has a better batting average compared to his team.

Use [link] to find the value that is three standard deviations:

• above the mean
• below the mean

Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84 .

Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84.

49.5–59.5 2
59.5–69.5 3
69.5–79.5 8
79.5–89.5 12
89.5–99.5 5
2. Daily Low Temperature Frequency
49.5–59.5 53
59.5–69.5 32
69.5–79.5 15
79.5–89.5 1
89.5–99.5 0
3. Points per Game Frequency
49.5–59.5 14
59.5–69.5 32
69.5–79.5 15
79.5–89.5 23
89.5–99.5 2
1. ${s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}=\sqrt{\frac{193157.45}{30}-{79.5}^{2}}=10.88$
2. ${s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}=\sqrt{\frac{380945.3}{101}-{60.94}^{2}}=7.62$
3. ${s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}=\sqrt{\frac{440051.5}{86}-{70.66}^{2}}=11.14$

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