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.
Page 112 How to Compute Quartiles1. 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).
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
Page 112 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.
Example 9 - QuartilesOrdered Cost of Ice Cream Bars (in dollars)
Box-and-Whisker Plots Five Number Summary: Lowest Value Q1 Median Q3 Highest Value The five number summary is used to create a graphic sketch of the data called a box-and- whisker plot.
Box-and-Whisker Plots 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.
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
Page 114How to Make a Box-and-Whisker Plot1. Find the five number summary2. Draw a vertical scale to include the lowest and highest data values3. To the right of the scale, draw a box from Q1 to Q34. Include a solid line through the box at the median level5. Draw vertical lines, called Box-and-Whisker Plot Figure 3-6 whiskers, from Q1 to the lowest value and from Q3 to the highest value.
Using the Calculator: Find the Five Number Summary Enter the data in L1 Hit STAT, tab over to CALC, choose 1: 1-VarStats Hit 2nd 1 ENTER Tab down to view the five number summary
Using the Calculator: Box-and-Whisker Plot1. Enter the data into L1 4. Highlight 1:Plot1, hit2. Hit WINDOW ENTER, make the Xmin=lowest value selections Xmax=highest value Xscl=1 Ymin=-5 Ymax=5 Yscl=1 5. Hit GRAPH 6. Press TRACE and tab3. Hit 2nd y= left/right to view values in the Five Number Summary