Listed are 32 ages for Academy Award winning best actors in order from smallest to largest.

18; 18; 21; 22; 25; 26; 27; 29; 30; 31; 31; 33; 36; 37; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77

1. Find the percentile of 37.
2. Find the percentile of 72.

Jesse was ranked 37^th in his graduating class of 180 students. At what percentile is Jesse’s ranking?

1. For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race?
2. The 20^th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20^th percentile in the context of the situation.
3. A bicyclist in the 90^th percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90^th percentile in the context of the situation.

1. For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race?
2. The 40^th percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40^th percentile in the context of the situation.

On an exam, would it be more desirable to earn a grade with a high or low percentile? Explain.

Mina is waiting in line at the Department of Motor Vehicles (DMV). Her wait time of 32 minutes is the 85^th percentile of wait times. Is that good or bad? Write a sentence interpreting the 85^th percentile in the context of this situation.

In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78^th percentile. Should Li be pleased or upset by this result? Explain.

In a study collecting data about the repair costs of damage to automobiles in a certain type of crash tests, a certain model of car had 💲1,700 in damage and was in the 90^th percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90^th percentile in the context of this problem.

The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system:

1. Students’ GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an “admissions index” score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12% of high school students in the state. In this context, what percentile does the top 12% represent?
2. Students whose GPAs are at or above the 96^th percentile of all students at their high school are eligible (called eligible in the local context), even if they are not in the top 12% of all students in the state. What percentage of students from each high school are “eligible in the local context”?

Suppose that you are buying a house. You and your realtor have determined that the most expensive house you can afford is the 34^th percentile. The 34^th percentile of housing prices is 💲240,000 in the town you want to move to. In this town, can you afford 34% of the houses or 66% of the houses?

First quartile = _______

Second quartile = median = 50^th percentile = _______

Third quartile = _______

Interquartile range (IQR) = _____ – _____ = _____

10^th percentile = _______

70^th percentile = _______

Use the following information to answer the next three exercises: Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Calculate the following:

mode = _______

Chapter Review from 2.6

Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are three types of distributions. A right (or positive) skewed distribution has a shape like [link]. A left (or negative) skewed distribution has a shape like [link]. A symmetrical distribution looks like [link].

Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right.

11122223333333344455

Solution

The data are symmetrical. The median is 3 and the mean is 2.85. They are close, and the mode lies close to the middle of the data, so the data are symmetrical.

161719222222222223

87878787878889899091

Solution

The data are skewed right. The median is 87.5 and the mean is 88.2. Even though they are close, the mode lies to the left of the middle of the data, and there are many more instances of 87 than any other number, so the data are skewed right.

When the data are skewed left, what is the typical relationship between the mean and median?

When the data are symmetrical, what is the typical relationship between the mean and median?

Solution

When the data are symmetrical, the mean and median are close or the same.

What word describes a distribution that has two modes?

Describe the shape of this distribution.

Solution

The distribution is skewed right because it looks pulled out to the right.

Describe the relationship between the mode and the median of this distribution.

Describe the relationship between the mean and the median of this distribution.

Solution

The mean is 4.1 and is slightly greater than the median, which is four.

Describe the shape of this distribution.

Describe the relationship between the mode and the median of this distribution.

Solution

The mode and the median are the same. In this case, they are both five.

Are the mean and the median the exact same in this distribution? Why or why not?

Describe the shape of this distribution.

Solution

The distribution is skewed left because it looks pulled out to the left.

Describe the relationship between the mode and the median of this distribution.

Describe the relationship between the mean and the median of this distribution.

Solution

The mean and the median are both six.

The mean and median for the data are the same.

345566667777777

Is the data perfectly symmetrical? Why or why not?

Which is the greatest, the mean, the mode, or the median of the data set?

111112121212131517222222

Solution

The mode is 12, the median is 13.5, and the mean is 15.1. The mean is the largest.

Which is the least, the mean, the mode, and the median of the data set?

5656565859606264646567

Of the three measures, which tends to reflect skewing the most, the mean, the mode, or the median? Why?

Solution

The mean tends to reflect skewing the most because it is affected the most by outliers.

In a perfectly symmetrical distribution, when would the mode be different from the mean and median?

Chapter Review from 2.7

The standard deviation can help you calculate the spread of data. There are different equations to use if you 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 = [latex]\sqrt{\frac{{\sum }^{\text{​}}{\left(x-\overline{x}\right)}^{2}}{n-1}}[/latex] or s = [latex]\sqrt{\frac{{\sum }^{\text{​}}f{\left(x-\overline{x}\right)}^{2}}{n-1}}[/latex] 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 = [latex]\sqrt{\frac{{\sum }^{\text{​}}{\left(x-\mu \right)}^{2}}{N}}[/latex] or σ = [latex]\sqrt{\frac{{\sum }^{\text{​}}f{\left(x-\mu \right)}^{2}}{N}}[/latex].

Formula Review

[latex]{s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}[/latex] where [latex]\begin{array}{l}{s}_{x}=\text{ sample standard deviation}\\ \overline{x}\text{ = sample mean}\end{array}[/latex]

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.

Solution

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

Solution

For Fredo: z = [latex]\frac{0.158\text{ – }0.166}{0.012}[/latex]
= –0.67

For Karl: z = [latex]\frac{0.177\text{ – }0.189}{0.015}[/latex]
= –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.

1. Grade	Frequency
   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

Solution

1. [latex]{s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}=\sqrt{\frac{193157.45}{30}-{79.5}^{2}}=10.88[/latex]
2. [latex]{s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}=\sqrt{\frac{380945.3}{101}-{60.94}^{2}}=7.62[/latex]
3. [latex]{s}_{x}=\sqrt{\frac{\sum f{m}^{2}}{n}-{\overline{x}}^{2}}=\sqrt{\frac{440051.5}{86}-{70.66}^{2}}=11.14[/latex]