Comparing Distributions

Comparing Measures of

Center and Variation

Choosing the most appropriate measures

Symmetric versus Skewed Data

Heights of Adults

Lengths of index fingers of 8th graders(mm)

Mean and Standard Deviation

Mean - the typical value (center) of the data when the distribution is roughly symmetric.

●The arithmetic average of the data

●Every data value impacts the mean

●Will be influenced by outliers

Standard Deviation - a measure of the typical distance of the observations (data) and the mean.

●Measures the spread of the distribution

●Measures the variability of a fairly symmetric distribution

●Will be influenced by outliers

Median and IQR

Median - the middle value in an ordered data set

●Roughly 50% of the data is below and above the median

●Measures the typical value for data in a skewed distribution

●NOT influenced by all data values

Interquartile Range (IQR) - the middle 50% of the data in a distribution

●Found by subtracting the 1st quartile (Q1) from the 3rd quartile (Q3)

●Measures the variability of a sample with a skewed distribution

Appropriate Measures of Center and

Variation

●

The shape of the distribution determines which measure of center
and variability are best.

●

Use the mean and standard deviation for distributions that are
unimodal and roughly symmetric.

●

Use the median and IQR for distributions with skewness or outliers.

●

The median and IQR are resistant to (unaffected by) outliers.

Comparing Measures of Center

●

In a symmetric distribution, the mean and the median are
approximately the same.

●

In a right skewed distribution the mean tends to be greater than the
median.

●

In a left skewed distribution the mean tends to be less than the
median.

Drag and Drop

This distribution is ​

Placeholder

so we should use the ​

Placeholder

as a measure of center and the ​

Placeholder

as a measure of variability.

Drag these tiles and drop them in the correct blank above

right skewed

median

IQR

fairly symmetric

left skewed

mean

standard deviation

range

mode

Drag and Drop

This distribution is ​

Placeholder

so we should use the ​

Placeholder

as a measure of center and the ​

Placeholder

as a measure of variability.

Drag these tiles and drop them in the correct blank above

fairly symmetric

mean

standard deviation

left skewed

range

mode

median

IQR

right skewed

Multiple Choice

Which measure of variation would we use with this data?

• Mean, since the data is fairly symmetric.

• Median, since the data is skewed.

• Standard deviation, since the data is symmetric.

• IQR, since the data is skewed.

Audio Response

How would you decide which measure of center to use with a set of data?

Open Audio Recorder

Draw

Please draw and label a boxplot with the following descriptive statistics:

Min= 5 Q1 = 7 Med=8 Q3 = 11 Max=15