Additional Measures of Center and Spread

Math AllianceFall 2011

Measures of CenterMode

Most frequent value

Mean

Fair share or balance point

Median

If odd number of values then middle number

If even number of values then mean of middle two numbers

Five Number SummaryMinimum: lowest value

1st Quartile: median of the lower half

Median: middle number

3rd Quartile: median of the upper half

Maximum: highest value

Box PlotGraph of the five number summary

Interpretation of Box PlotsThe five number summary divides a data distribution into 4 parts. About what percent of the data values in

each of the following intervals?

• before the median

• after the median

• in the box (between the 1st and 3rd quartiles)

• before the upper quartile

• after the upper quartile

• before the lower quartile

• after the lower quartile

• between median and upper quartile

• between the median and the lower quartile

Definition of outliersSteps to determine if there are outliers:

1. Find the Interquartile range (IQR)

IQR = Q3 – Q1

2. Multiply: 1.5 * IQR

3. Add: Q3 + 1.5*IQR

4. Any value greater than Q3+ 1.5*IQR is an outlier

5. Subtract Q1 – 1.5*IQR

6. Any value less than Q1-1.5*IQR is an outlier

Definition of OutlierAny value more than 1.5 IQRs above Q3 or below Q1.

Or

Any value more than 1.5 “boxes” above Q3 or below Q1

Example: Natural Peanut Butter Quality Ratings:

34 40 52 57 57 60 60 63 67 69 69 69 71 89

Find the 5 number summary

Make a box plot

Determine if there are any outliers

Comparing two or more Groups

Side by side box plots

Calories for all beef hot dogs:

157 149 131 111 149 152 190 184 175 190 139 181 148 176 158 132 141 186 135 153

Calories for all poultry hot dogs:

170 152 146 142 102 135 94 106 86 113 102 143 99 132 144 129 87

Comparing two groupsEach each type of hot dog

1.Find the 5 number summary

2.Construct a box plot of each using the same scale for both. Place the beef hot dog box plot above the poultry hot dog box plot.

3.What type of hot dog has the fewer number of calories? Use the box plots and the percent to explain your answer.