CHAPTER 3AVERAGES AND VARIATION

3.3 Percentiles and Box-and-Whisker Plots

Percentiles and Box-and-Whisker Plots

We’ve seen measures of central tendency and spread for a set of data. The arithmetic mean and the standard deviation s will be very useful in later work. However, because they each utilize every data value, they can be heavily influenced by one or two extreme data values.

In cases where our data distributions are heavily skewed or even bimodal, we often get a better summary of the distribution by utilizing relative position of data rather than exact values.

Percentiles

For whole numbers P (where 1 ≤ P ≤ 99), the Pth percentile of a distribution is a value such that P% of the data fall at or below it and (100 – P)% of the data fall at or above it. There are 99 percentiles, and in an ideal situation, the

99 percentiles divide the data set into 100 equal parts. However, if the number of data elements is not exactly divisible by 100, the percentiles will not divide the data into equal parts.

Finding Percentiles

There are several widely used conventions for finding percentiles. They lead to slightly different values for different situations, but these values are close together.

To find percentiles:1. Order the values from smallest to largest.2. A natural way to find the Pth percentile is to then find a value

such that P% of the data fall at or below it. This will not always be possible, so we take the nearest value satisfying the criterion. It is at this point that there is a variety of processes to determine the exact value of the percentile. We will not be very concerned about exact procedures for evaluating percentiles in general.

Quartiles

Quartiles are those percentiles that divide the data into fourths. The first quartile Q1 is the 25th percentile, the second quartile Q2 is the median and the 50th percentile, and the third quartile Q3 is the 75th percentile.

How to Compute Quartiles

1. Order the data from smallest to largest2. Find the median. This is Q2.3. The first quartile Q1 is then the median of

the lower half of the data; that is it is the median of the data falling below the Q2 position (and not including Q2).

4. The third quartile Q3 is then the median of the upper half of the data; that is it is the median of the data falling above the Q2 position (and not including Q2).

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Interquartile Range

The interquartile range (IQR) is the difference between the third and first quartiles. Tells the spread of the middle half of the data Resistant to change because extreme values would be

found in the lower and upper quartiles

IQR = Q3 – Q1

Example 9 - Quartiles

In a hurry? On the run? Hungry as well? How about an ice cream bar as a snack? Ice cream bars are popular among all age groups. Consumer Reports did a study of ice cream bars. Twenty-seven bars with taste ratings of at least “fair” were listed, and cost per bar was included in the report. Just how much does an ice cream bar cost?

(a) Find the quartiles.(b) Find the interquartile range.

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Example 9 - Quartiles

Ordered Cost of Ice Cream Bars (in dollars)

Box-and-Whisker Plots

Five Number Summary:Lowest ValueQ1MedianQ3Highest Value

The five number summary is used to create a graphic sketch of the data called a box-and-whisker plot.

A box-and-whisker plot is a visual representation a five-number summary. provides a graphic display of the spread of the data

about the median and a quick overall summary of symmetry/skewness The box shows the middle half of the data. One quarter of

the data is located along each whisker. If the median line is near one end of the box, the data

are skewed toward the other end of the box. If the median is approximately centered in the box

and the whiskers are about the same length, the data is approximately symmetric.

Box-and-Whisker Plots

Outliers

How far away is “far enough” to be considered an outlier? Any number that is beyond 1.5(IQR) from either Q1 or

Q3 is considered an outlier

How to Make a Box-and-Whisker Plot

1. Find the five number summary2. Draw a vertical scale to

include the lowest and highest data values

3. To the right of the scale, draw a box from Q1 to Q3

4. Include a solid line through the box at the median level

5. Draw vertical lines, called whiskers, from Q1 to the lowest value and from Q3 to the highest value.

Box-and-Whisker Plot

Figure 3-6

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Using the Calculator: Find the Five Number Summary

Enter the data in L1Hit STAT, tab over to CALC, choose 1: 1-

VarStatsHit 2nd 1 ENTERTab down to view the five number summary

Using the Calculator:Box-and-Whisker Plot

1. Enter the data into L1

2. Hit WINDOWXmin=lowest valueXmax=highest valueXscl=1Ymin=-5Ymax=5Yscl=1

3. Hit 2nd y=

4. Highlight 1:Plot1, hit ENTER, make the selections

5. Hit GRAPH6. Press TRACE and tab

left/right to view values in the Five Number Summary

Assignment

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