Section 2.5
Summer 2013 - Math 1040
June 10, 2013
(1040)
M 1040 - 2.5
June 10, 2013
1 / 14
Roadmap
We will study today:
§2.5 Measures of Position.
Quartiles.
Boxplots.
Standard scores (z-scores).
(1040)
M 1040 - 2.5
June 10, 2013
2 / 14
Quantiles and quartiles
Quantiles are numbers that divide an ordered data set into lower and
upper percentages. For instance, the median is known as a 50-percentile
because it divides the data into a lower 50% and upper 50%.
(1040)
M 1040 - 2.5
June 10, 2013
3 / 14
Quantiles and quartiles
Quantiles are numbers that divide an ordered data set into lower and
upper percentages. For instance, the median is known as a 50-percentile
because it divides the data into a lower 50% and upper 50%.
Quartiles divide the data into quarters. We can label them Q1 , Q2 , Q3 ,
respectively dividing the data into the lower 25%, 50%, and 75%. That is,
about one-quarter of the data falls below Q1 , two-quarters below Q2 , and
three-quarters below Q3 .
(1040)
M 1040 - 2.5
June 10, 2013
3 / 14
Quartiles
Data set: 15 CPR training test scores are below.
13 9 18 15 14 21 7 10 11 20 5 18 37 16 17
are first sorted sorted
5 7 9 10 11 13 14 15 16 17 18 18 20 21 27
The median 15 divides the data into two equal parts.
5 7 9 10 11 13 14 15 16 17 18 18 20 21 27
Lower half: 5 7 9 10 11 13 14
Upper half: 16 17 18 18 20 21 27
(1040)
M 1040 - 2.5
June 10, 2013
4 / 14
Finding quartiles
1. Sort the data.
2. Find the median Q2 .
3. Find the medians of the lower and upper halves. These are Q1 and Q3 .
Special rule: When the sample size is even, average the middle values.
(1040)
M 1040 - 2.5
June 10, 2013
5 / 14
IQR
We recall the range of a data set is its maximum minus the minimum.
The interquartile range (IQR) is the range of the middle 50% of the
data. That is,
IQR = Q3 − Q1 .
(1040)
M 1040 - 2.5
June 10, 2013
6 / 14
IQR
We recall the range of a data set is its maximum minus the minimum.
The interquartile range (IQR) is the range of the middle 50% of the
data. That is,
IQR = Q3 − Q1 .
The IQR of the CPR training scores is 18 − 10 = 8. The spread of the
middle part of the data is 8 test points.
(1040)
M 1040 - 2.5
June 10, 2013
6 / 14
IQR
Example Tail lengths of a sample of American alligators, in feet:
6.5 3.4 4.2 7.1 5.4 6.8 7.5 3.9 4.6
1. Sort the data.
3.4 3.9 4.2 4.6 5.4 6.5 6.8 7.1 7.5
2. Find the median Q2 .
3.4 3.9 4.2 4.6 5.4 6.5 6.8 7.1 7.5
Q2 = 5.4 feet.
3. Find Q1 and Q3 .
3.4 3.9 (4.05) 4.2 4.6 and 6.5 6.8 (6.95) 7.1 7.5
Q1 = 4.05 feet and Q3 = 6.95 feet with the special rule.
The IQR is then 6.95 - 4.05 = 2.9 feet.
(1040)
M 1040 - 2.5
June 10, 2013
7 / 14
Example
A question is asked to 10 people in a survey. Given the age of a Male, what
is the acceptable minimum age for dating a Female? Using R’s built-in
data set based on age-difference in dating, the following table is obtained,
giving the frequency of that amount. For instance, for a Male of age 21,
eight people believe the acceptable minimum age for a Female date is 17.
(1040)
M 1040 - 2.5
June 10, 2013
8 / 14
Example
Out of the 10 people surveyed, here is are the responses for a Male that is
age 30:
18 19 21 22 23 25 25 25 25 25.
(1040)
M 1040 - 2.5
June 10, 2013
9 / 14
Example
Out of the 10 people surveyed, here is are the responses for a Male that is
age 30:
18 19 21 22 23 25 25 25 25 25.
The statistics are:
Q1 = 21 years, Q2 = 24 years, Q3 = 25 years. Also, x̄ = 22.8 years.
Notice here that 25 is the mode. Do you believe this complicates things?
How do you interpret Q3 ?
(1040)
M 1040 - 2.5
June 10, 2013
9 / 14
Example
It turns out that in this example we can speak of these data in terms of
better divisions - percentiles. For instance, ask yourself what percent of the
ages are less than or equal to 26 years, 25 years, 24 years, 23 years, etc?
Less than or equal to: Percentage:
Percentile:
25 years
100%
100th-percentile
23 years
50%
50th-percentile
22 years
40%
40th-percentile
21 years
30%
30th-percentile
19 years
20%
20th-percentile
18 years
10%
10th-percentile
(1040)
M 1040 - 2.5
June 10, 2013
10 / 14
Box plots
Box-and-Whisker Plots
Box-and-whisker plots, or box-plots, are graphical representations of the
minimum, quartiles, and maximum (the so-called five number summary).
1. Find the five-number summary.
2. Construct a horizotal scale that spans the range.
3. Plot the five-number summary above the horizontal scale.
4. Draw a box above the scale from Q1 to Q3 , and draw a vertical line at
Q2 inside the box.
5. Draw whiskers from the box to the minimum and maximum values.
(1040)
M 1040 - 2.5
June 10, 2013
11 / 14
Standard scores
Standard Scores
The standard score or z-score is the number (possibly fractional) of
standard deviations a data value is from the mean. The formula is given
by:
x −µ
Value - Mean
=
Standard Deviation
σ
In the formula above it is more useful to memorize the middle, verbal part.
z=
(1040)
M 1040 - 2.5
June 10, 2013
12 / 14
Standard scores
A statistics test has a mean of µ1 = 63 and a standard deviation of
σ1 = 7.0 and a biology test has a mean of µ2 = 23 and standard deviation
of σ2 = 3.9.
(1040)
M 1040 - 2.5
June 10, 2013
13 / 14
Standard scores
A statistics test has a mean of µ1 = 63 and a standard deviation of
σ1 = 7.0 and a biology test has a mean of µ2 = 23 and standard deviation
of σ2 = 3.9.
A student gets a 75 on a statistics test and a 25 on a biology test. On
which test is the better score?
(1040)
M 1040 - 2.5
June 10, 2013
13 / 14
Standard scores
A statistics test has a mean of µ1 = 63 and a standard deviation of
σ1 = 7.0 and a biology test has a mean of µ2 = 23 and standard deviation
of σ2 = 3.9.
A student gets a 75 on a statistics test and a 25 on a biology test. On
which test is the better score?
(1040)
z1 =
75 − 63
≈ 1.71
7.0
z2 =
25 − 23
≈ 0.51
3.9
M 1040 - 2.5
June 10, 2013
13 / 14
Assignements
Assignment:
1
Read pages 100 - 106.
2
Exercises p 107, 1 - 49 odd.
Vocabulary: Quartiles, box plots, z-scores.
Understand: Use the median of a data set to help find quartiles.
Construct a box plot from the data set. Find the z-scores of data.
(1040)
M 1040 - 2.5
June 10, 2013
14 / 14