Understandable
Statistics
Chapter 3
Averages
And
Variations
Section 3
Percentiles
and
Box-and-Whisker Plots
Vocabulary
Box-and-Whiskers plot
Inter-quartile range
Outliers
Percentiles
Quartiles
Q1
Q2
Q3
Percentiles and Box-and-Whisker Plots
Percentiles
★
For any whole number P (between 1 and
99), the Pth percentile of a distribution is a
value such that P% of the data fall at or
below it.
★
The percent falling above the Pth percentile
will be (100 – P)%.
Percentiles and Box-and-Whisker Plots
Percentiles
P40
Highest Value
Lowest Value
}
40% of data
}
60% of data
Determine the value for the
40th percentile of the given data
Data:
72
87
83
63
97 100
87 93
39
87
45
79
Steps to find the percentile:
1. Arrange the data from largest to smallest.
2. Multiple the percentile by the number of data
items.
3. Round the number down to closest integer value.
4. Count that number of data items up form the
bottom of the list.
76
85
Determine the value for the
40th percentile of the given data
Data:
72
87
83
63
97 100
87 93
39
87
45
79
76
85
Steps to find the percentile:
1. 100, 97, 93, 87, 87, 87, 85, 83, 79, 76, 72, 63, 45, and 39
2. P40 = 0.40 x 14
P40 = 5.6
3. P40 = 5, since 5.6 rounds down to 5
4. The data value that would represent the 40th percentile
is 76. 40% of the data is at or below this value.
Determine the value for the
75th percentile of the given data
Data:
72
87
83
63
97 100
87 93
39
87
45
79
76
85
Steps to find the percentile:
1. 100, 97, 93, 87, 87, 87, 85, 83, 79, 76, 72, 63, 45, and 39
2. P75 = 0.75 x 14 = 10.5
P75 = 10.5
3. P75 = 10, since 10.5 rounds down to 10
4. The data value that would represent the 75th percentile
is 87. 75% of the data is at or below this value.
Percentiles and Box-and-Whisker Plots
Quartiles
✤Percentiles that divide the data into fourths
✤Q1 = 25th percentile
✤Q2 = the median
✤Q3 = 75th percentile
Percentiles and Box-and-Whisker Plots
Q1
Median = Q2
Q3
Inter-quartile range = IQR = Q3 — Q1
Highest Value
Lowest Value
Quartiles
Percentiles and Box-and-Whisker Plots
Computing Quartiles
✦ Order the data from smallest to largest.
✦ Find the median, the second quartile.
✦ Find the median of the data falling below Q2. This is the
first quartile.
✦ Find the median of the data falling above Q2. This is the
third quartile.
Percentiles and Box-and-Whisker Plots
Find the quartiles:
12
15
16
16
17
18
22
22
23
24
25
30
32
33
33
34
41
45
58
The median is
24
Percentiles and Box-and-Whisker Plots
Find the quartiles:
12
15
16
16
17
18
22
22
23
24
25
30
32
33
33
34
41
45
58
For the data below the median, the median is
So, Q1 = 17.
17 .
Percentiles and Box-and-Whisker Plots
Find the quartiles:
12
15
16
16
17
18
22
22
23
24
25
30
32
33
33
34
41
45
58
For the data below the median, the median is
So, Q3 = 33.
33.
Percentiles and Box-and-Whisker Plots
Find the interquartile range:
12
15
16
16
17
18
22
22
23
24
25
30
32
33
33
34
41
45
58
IQR = Q3 – Q1
IQR = 33 – 17
IQR = 16
Percentiles and Box-and-Whisker Plots
Five-Number
Summary of Data
✓ Lowest value
✓ First quartile
✓ Median
✓ Third quartile
✓ Highest value
12
17
24
33
58
Percentiles and Box-and-Whisker Plots
Box-and-Whisker Plot
A graphical presentation of the
five-number summary of data.
Percentiles and Box-and-Whisker Plots
Making a Box-and-Whisker Plot
✦ Draw a vertical scale including the lowest and highest
values.
✦ To the right of the scale, draw a box from Q1 to Q3.
✦ Draw a solid line through the box at the median.
✦ Draw lines (whiskers) from Q1 to the lowest and from
Q3 to the highest values.
Percentiles and Box-and-Whisker Plots
Box-and-Whisker Plot
Highest = 58
Lowest = 12
Q1 = 17
10
Median = 24
20
Q3 = 33
30
40
50
60
Percentiles and Box-and-Whisker Plots
Outliers
Outlier < Q1 − 1.5IQR
Outlier < 17 − 1.5 (16 )
Outlier > Q3 + 1.5IQR
Outlier > 33 + 1.5 (16 )
Outlier < 17 − 24
Outlier < −7
Outlier > 33 + 24
Outlier > 57
Outliers for the given data set: 58
Percentiles and Box-and-Whisker Plots
Box-and-Whisker Plot with
Outliers
*
Lowest = 12
Q1 = 17 Median = 24
10
20
Q3 = 33
30
Largest data item
that is NOT an
outlier = 45
40
50
Highest = 58
(Outlier)
60
The
End