MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

3.1: SUMMARIES FOR SYMMETRIC DISTRIBUTIONS

In chapter two we learned that we can characterize the typical value of a distribution by the center of the distribution and the variability in that distribution by the horizontal spread. We left these concepts somewhat vague, but now our goal is to quantify them (i.e. measure them with numbers).

In this section we will investigate the mean and standard deviation for symmetric distributions.

The Center as the Balancing Point: The Mean

The mean of a collection of data is the arithmetic average. We can think of the mean as the “balancing point of the distribution”.

Visualizing the Mean

The mean for a symmetric distribution is roughly in the middle. However, the mean for a skewed distribution may appear off-center.

Example 1 – Visualizing the Mean in Histograms

The figures below show two distributions: one with ACT scores and one with salary (in millions of $).

a. Which distribution appears to be symmetric?

Approximately what value is the mean? What do we notice about the mean of that distribution?

b. Which distribution appears to be skewed?

Approximately what value is the mean? What do we notice about the mean of that distribution?

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MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

The Mean in Context

The mean of a collection of data is located at the “balancing point” of a distribution of data. The mean is one representation of the “typical” value of a variable.

Calculating the Mean

The arithmetic mean (average), x (x-bar), is the sum of all of the variables in the data set divided by the number of observations.

Formula :Mean=x=∑ xn

x=mean ,∑ ¿ summation , x=indiviudal data points , n=sample¿¿

Example 2 – Gas Prices

According to GasBuddy.com, the prices of 1 gallon of regular gas at 12 service stations in a neighborhood in Austin, Texas, were as follows on one fall day in 2013:

$3.19, $3.09, $3.09, $2.93, $2.95, $3.09, $2.99, $2.99, $2.95, $2.99, $2.99, $2.97

a. Find the mean price of a gallon of regular gas at these service stations. (Note: The distribution is fairly symmetric if plotted.)

x=∑ xn

=❑=¿

b. Explain what the value means in context (i.e. interpret the mean).

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MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

STATCRUNCH – MEAN

- Choose one of the options below:o Sign in to MyStatLab & select StatCrunch on the left side of the screen.

- Type or paste data into a blank data table OR - view the data sets from your textbook in StatCrunch & choose the problem number

o Sign in to statcrunch.com directly and enter your MyStatLab login information.- Enter data values into a column in the table.- Go to “Stat”- Select “Summary Stat”- Select “Column”- Select the column at the top of the window.- “Compute” *NOTE: This is the last time the first step will be included in the StatCrunch steps, because it is always the same.*

Example 2 – Gas Prices – Revisited

According to GasBuddy.com, the prices of 1 gallon of regular gas at 12 service stations in a neighborhood in Austin, Texas, were as follows on one fall day in 2013:

$3.19, $3.09, $3.09, $2.93, $2.95, $3.09, $2.99, $2.99, $2.95, $2.99, $2.99, $2.97

Find the mean price of a gallon of regular gas at these service stations using StatCrunch.

Summary statistics:Column n Mean Variance Std. dev. Std. err. Median Range Min Max Q1 Q3

Gas Prices 12 3.0183333 0.0061424242 0.07837362 0.022624515 2.99 0.26 2.93 3.19 2.96 3.09Note: StatCrunch gives you a lot of information. We will discuss more soon.

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MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

Standard Deviation

The standard deviation is a number that measures how far away the typical observation is from the mean.

Visualizing the Standard Deviation

The following histograms record the daily high temperatures in degrees Fahrenheit over one recent year at two locations: Provo, Utah and San Francisco, California.

Both distributions have roughly the same shape (symmetric) and mean (67° in Provo and 65° in San Francisco), although the variations in temperature is much greater in Provo than in San Francisco.

Example 3 – Visualizing Standard Deviation in Histograms

The figures below show histograms for distributions of the same number of distributions, and all of the distributions have a mean value of about 3.5.

Based on the histograms,

a. which distribution has the largest standard deviation Why?

b. which has the smallest standard deviation? Why?

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MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

The Standard Deviation in Context

The standard deviation is more abstract than the mean. In a symmetric, unimodal distribution, the majority of the observations (about two-thirds) are less than one standard deviation from the mean.

Calculating the Standard Deviation

The standard deviation, s, is the square root of the sum of all of the deviations from the mean in the data set divided by the number of observations.

Formula :Standard Deviation=s=√∑ ( x−x )2

n−1

s=standard deviation ,∑ ¿ summation , x=indiviudal data points , x=mean,n=sample¿ ( x−x )=deviation

Example 4 – Calculating Standard Deviation

Find the standard deviation of the following data set:

6, 8, 9, 10, 12, 15

a. Find the standard deviation.

x=∑ xn

=¿

s=√∑ ( x−x )2

n−1=¿

b. Explain what the value means in context (i.e. interpret the standard deviation).

STATCRUNCH – STANDARD DEVIATION

- Enter data values into a column in the table.

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Step 1

Step 3Step 2

Step 5

Step 4

x x−x ( x−x )2

6

8

9

10

12

15

∑ (x−x )2 =

MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

- Go to “Stat”- Select “Summary Stat”- Select “Column”- Select the column at the top of the window.- “Compute”

Example 2 – Gas Prices – Revisited

According to GasBuddy.com, the prices of 1 gallon of regular gas at 12 service stations in a neighborhood in Austin, Texas, were as follows on one fall day in 2013:

$3.19, $3.09, $3.09, $2.93, $2.95, $3.09, $2.99, $2.99, $2.95, $2.99, $2.99, $2.97

a. Find the standard deviation of the price of a gallon of regular gas at these service stations using Statcrunch.

Summary statistics:Column n Mean Variance Std. dev. Std. err. Median Range Min Max Q1 Q3

Gas Prices 12 3.0183333 0.0061424242 0.07837362 0.022624515 2.99 0.26 2.93 3.19 2.96 3.09Note: StatCrunch gives you a lot of information. We will discuss more soon.

b. Explain what the value means in context (i.e. interpret the standard deviation).

Example 5 – Guitar Lessons

Compare the following lists of two different groups of first-year guitar lesson students’ ages.

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MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

Group 1 Ages: 9, 9, 10, 10, 10, 12 Group 2 Ages: 7, 8, 10, 11, 11, 13

a. Find the mean of each group.

Group 1: x=∑ xn

=¿ Group 2: x=∑ xn

=¿

b. Guess which of the following list will have a larger standard deviation. Why?

c. Find the standard deviation of each group (using StatCrunch).

Group 1: s=¿ Group 2: s=¿

Does this support your answer to part b?

Example 6 – Smog Levels

The graph shows the distribution of the amount of particulate matter, or smog, in the air in 333 cities in the United States in 2008, as reported by the Environmental Protection Agency (EPA). The mean particulate matter is 10.7 micrograms per cubic meter, and the standard deviation is 2.6 micrograms per cubic meter.

a. Find the level of particulate matter one standard deviation above and below the mean.

b. Keeping in mind that the EPA says that levels of over 15 micrograms per cubic meter are unsafe, what can we conclude about the air quality of most of the cities in this sample?

Example 7 – Children’s Ages

Mrs. Johnson’s children are 2, 2, 3, and 5 years of age.

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MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

a. Find the mean and standard deviation using StatCrunch.

Summary statistics:

b. Find the mean and standard deviation in 20 years using StatCrunch.(We will be using these four ages: ____, ____, ____, ____)

Summary statistics:

a. Is the mean of their ages in 20 years larger, smaller, or the same? Why?

b. Is the standard deviation of their ages in 20 years larger, smaller, or the same? Why?

Variance

Another way of measuring spread is the variance. Variance is closely related to standard deviation, it is the variation squared.

Formula :Variance=s2=∑ ( x−x )2

n−1

s=standard deviation ,∑ ¿ summation , x=indiviudal data points , x=mean,n=sample¿ ( x−x )=deviation

For most applications, the standard deviation is preferred over the variance since it has the same units as the original data set. (Notice the variance column next to the standard deviation column in the StatCrunch tables in examples 2 and 3.)

Example 8 – Travel Times (In-Class Practice)

The following data represent the travel times (in minutes) to work for all seven employees of a web-development company: 23, 36, 23, 18, 5, 26, 43

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Column n Mean Std. dev.

Current Age

Column n Mean Std. dev.

Current Age

MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

a. Find the mean of the travel times. (Note: The distribution is fairly symmetric if plotted.)

x=∑ xn

=¿

b. Verify the mean you found above and find the standard deviation using StatCrunch.

Column n Mean Variance Std. dev. Std. err. Median Range Min Max Q1 Q3

Gas Prices 7 150.47619 4.6364425 23 38 5 43 18 36

c. Explain what the mean value means in context (i.e. interpret the mean).

d. Explain what the standard deviation value means in context (i.e. interpret the standard deviation).

Snapshot: The Mean of a Sample

What is it? A numerical summary.

What does it do? Measures the center of the distribution of a sample of data.

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MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 3.1

How does it do it?

The mean identifies the “balancing point” of the distribution, which is the arithmetic average of the values.

How is it used? The mean represents the typical value in a set of data when the distribution is roughly symmetric.

Snapshot: The Standard Deviation of a Sample

What is it? A numerical summary.

What does it do? Measures the spread of a distribution of a sample of data.How does it do it? It measures the typical distance from the mean.

How is it used? To measure the amount of variability in a sample when the distribution is fairly symmetric.

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