§3.1–Measures of Center

Tom Lewis

Fall Term 2009

Tom Lewis () §3.1–Measures of Center Fall Term 2009 1 / 8

Outline

1 The mean

2 The median

3 The mode

4 Comparisons of measures of center

5 The sample mean

Tom Lewis () §3.1–Measures of Center Fall Term 2009 2 / 8

The mean

Definition (The mean)

The mean of a data set is the sum of the observations divided by thenumber of observations. In other words, it is the average value of the dataset.

Problem

Compute the mean of the ACT data. Use the three different methods:

Calculator method 1

Calculator method 2

R

Tom Lewis () §3.1–Measures of Center Fall Term 2009 3 / 8

The mean

Definition (The mean)

The mean of a data set is the sum of the observations divided by thenumber of observations. In other words, it is the average value of the dataset.

Problem

Compute the mean of the ACT data. Use the three different methods:

Calculator method 1

Calculator method 2

R

Tom Lewis () §3.1–Measures of Center Fall Term 2009 3 / 8

The mean

Definition (The mean)

The mean of a data set is the sum of the observations divided by thenumber of observations. In other words, it is the average value of the dataset.

Problem

Compute the mean of the ACT data. Use the three different methods:

Calculator method 1

Calculator method 2

R

Tom Lewis () §3.1–Measures of Center Fall Term 2009 3 / 8

The mean

Definition (The mean)

The mean of a data set is the sum of the observations divided by thenumber of observations. In other words, it is the average value of the dataset.

Problem

Compute the mean of the ACT data. Use the three different methods:

Calculator method 1

Calculator method 2

R

Tom Lewis () §3.1–Measures of Center Fall Term 2009 3 / 8

The mean

Definition (The mean)

The mean of a data set is the sum of the observations divided by thenumber of observations. In other words, it is the average value of the dataset.

Problem

Compute the mean of the ACT data. Use the three different methods:

Calculator method 1

Calculator method 2

R

Tom Lewis () §3.1–Measures of Center Fall Term 2009 3 / 8

The median

Definition (The median)

The median of a data set is a number that divides the data set in half:half the data are below the median and half the data are above themedian. For a set of n values, it is calculated as follows:

Arrange the data in increasing order.

If n is odd, then the median is the observation in the middle of theordered list, at position (n + 1)/2.

If n is even, then the median is the average of the two middleobservations in the middle of the ordered list.

Problem

Find the median of the ACT data.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 4 / 8

The median

Definition (The median)

The median of a data set is a number that divides the data set in half:half the data are below the median and half the data are above themedian. For a set of n values, it is calculated as follows:

Arrange the data in increasing order.

If n is odd, then the median is the observation in the middle of theordered list, at position (n + 1)/2.

If n is even, then the median is the average of the two middleobservations in the middle of the ordered list.

Problem

Find the median of the ACT data.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 4 / 8

The median

Definition (The median)

The median of a data set is a number that divides the data set in half:half the data are below the median and half the data are above themedian. For a set of n values, it is calculated as follows:

Arrange the data in increasing order.

If n is odd, then the median is the observation in the middle of theordered list, at position (n + 1)/2.

If n is even, then the median is the average of the two middleobservations in the middle of the ordered list.

Problem

Find the median of the ACT data.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 4 / 8

The median

Definition (The median)

The median of a data set is a number that divides the data set in half:half the data are below the median and half the data are above themedian. For a set of n values, it is calculated as follows:

Arrange the data in increasing order.

If n is odd, then the median is the observation in the middle of theordered list, at position (n + 1)/2.

If n is even, then the median is the average of the two middleobservations in the middle of the ordered list.

Problem

Find the median of the ACT data.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 4 / 8

The median

Definition (The median)

The median of a data set is a number that divides the data set in half:half the data are below the median and half the data are above themedian. For a set of n values, it is calculated as follows:

Arrange the data in increasing order.

If n is odd, then the median is the observation in the middle of theordered list, at position (n + 1)/2.

If n is even, then the median is the average of the two middleobservations in the middle of the ordered list.

Problem

Find the median of the ACT data.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 4 / 8

The mode

Definition (The mode)

The mode of a data set is the most frequent observation in the data set.Please note:

If no value occurs more than once, then the data set has no mode.

Otherwise, any value that occurs with greatest frequency is a mode ofthe data set.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 5 / 8

The mode

Definition (The mode)

The mode of a data set is the most frequent observation in the data set.Please note:

If no value occurs more than once, then the data set has no mode.

Otherwise, any value that occurs with greatest frequency is a mode ofthe data set.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 5 / 8

The mode

Definition (The mode)

The mode of a data set is the most frequent observation in the data set.Please note:

If no value occurs more than once, then the data set has no mode.

Otherwise, any value that occurs with greatest frequency is a mode ofthe data set.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 5 / 8

Comparisons of measures of center

Problem

Find the mode of the ACT data.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 6 / 8

Comparisons of measures of center

Comparisons of measures of center

How can we compare the mean, median and mode of a data set? Whatmight we expect for data sets with the following distribution shapes?

Bell

Right-skewed

Left-skewed

Uniform

J shaped

Tom Lewis () §3.1–Measures of Center Fall Term 2009 6 / 8

Comparisons of measures of center

Comparisons of measures of center

How can we compare the mean, median and mode of a data set? Whatmight we expect for data sets with the following distribution shapes?

Bell

Right-skewed

Left-skewed

Uniform

J shaped

Tom Lewis () §3.1–Measures of Center Fall Term 2009 6 / 8

Comparisons of measures of center

Comparisons of measures of center

How can we compare the mean, median and mode of a data set? Whatmight we expect for data sets with the following distribution shapes?

Bell

Right-skewed

Left-skewed

Uniform

J shaped

Tom Lewis () §3.1–Measures of Center Fall Term 2009 6 / 8

Comparisons of measures of center

Comparisons of measures of center

How can we compare the mean, median and mode of a data set? Whatmight we expect for data sets with the following distribution shapes?

Bell

Right-skewed

Left-skewed

Uniform

J shaped

Tom Lewis () §3.1–Measures of Center Fall Term 2009 6 / 8

Comparisons of measures of center

Comparisons of measures of center

How can we compare the mean, median and mode of a data set? Whatmight we expect for data sets with the following distribution shapes?

Bell

Right-skewed

Left-skewed

Uniform

J shaped

Tom Lewis () §3.1–Measures of Center Fall Term 2009 6 / 8

Comparisons of measures of center

Comparisons of measures of center

How can we compare the mean, median and mode of a data set? Whatmight we expect for data sets with the following distribution shapes?

Bell

Right-skewed

Left-skewed

Uniform

J shaped

Tom Lewis () §3.1–Measures of Center Fall Term 2009 6 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.

It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.

It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.

Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

Comparisons of measures of center

A summary of measures of center

Mean

This is a standard measure of center for quantitativedata.It can be influenced by outlying values (non-resistant).

Median

The median resists the influence of outliers.It does not always seem to be a faithful measure of thecenter.

Mode

Especially helpful for qualitative data.Can be a misleading measure if the data set is small.

Tom Lewis () §3.1–Measures of Center Fall Term 2009 7 / 8

The sample mean

Definition (The sample mean)

For a variable x , the mean of the observations for a sample is called asample mean and is denoted by x . If the sample data is {x1, x2, . . . , xn},then

x =sum of data

number of observations=

∑ni=1 xi

n=

∑xi

n

Tom Lewis () §3.1–Measures of Center Fall Term 2009 8 / 8