7- 1
Chapter 3
Measures of Position
Seven edition - Elementary statistic
Bluman
7- 2
Measures of Position
1. Z-score
2. Percentile
3. Quartile
4. Outlier
7- 3
Introduction
In addition to measures of central tendency and
measures of variation, there are measures of
position or location. These measures include
standard scores, percentiles, deciles, and quartiles.
They are used to locate the relative position of a data
value in the data set.
7- 4
Introduction(cont’d)
Examples
If a value is located at the 80th percentile, it means that
80% of the values fall below it in the distribution and
20% of the values fall above it.
The median is the value that corresponds to the 50th
percentile, since one-half of the values fall below it and
one-half of the values fall above it.
7- 5
Z-score or standard score
•
7- 6
7- 7
7- 8
Note that if the z score is positive, the score is above the
mean. If the z score is 0, the score is the same as the
mean. And if the z score is negative, the score is below
the mean.
When all data for a variable are transformed into z
scores, the resulting distribution will have a mean of 0
and a standard deviation of 1.
7- 9
Percentiles
7- 10The class
7- 11
Finding a Data Value Corresponding to a
Given Percentile
7- 12
Finding a Data Value Corresponding to
a Given Percentile
Step 3
If c is not a whole number, round it up to the next whole
number; in this case, c = 3.Start at the lowest value and
count over to the third value, which is 5. Hence, the value
5 corresponds to the 25th percentile.
7- 13
7- 14
2, 3, 5, 6, 8, 10, 12, 15, 18, 20
6th value 7th value
The value halfway between 10 and 12 is 11. Find it by
adding the two values and dividing by 2.
Hence, 11 corresponds to the 60th percentile. Anyone
scoring 11 would have done better than 60% of the class.
7- 15
Quartiles and Deciles
Quartiles: divide the distribution into four groups,
separated by Q1, Q2, Q3.
Note that :
Q1 is the same as the 25th percentile;
Q2 is the same as the 50th percentile, or the median;
Q3 corresponds to the 75th percentile, as shown
7- 16
Calculate the percentile
Finding Data Values Corresponding to Q1, Q2, and Q3
Step 1 Arrange the data in order from lowest to highest.
Step 2 Find the median of the data values. This is the value
for Q2.
Step 3 Find the median of the data values that fall below
Q2. This is the value for Q1.
Step 4 Find the median of the data values that fall above
Q2. This is the value for Q3.
7- 17
Quartiles is found in this procedure table
Procedure Table
Finding Data Values Corresponding to Q1, Q2, and Q3
Step1 Arrange the data in order from lowest to highest
Step2 Find the median of the data values. This is the value for Q2.
Step3 Find the median of the data values that fall below Q2. This is the value for Q1.
Step4 Find the median of the data values that fall above Q2. This is the value for Q3.
7- 18
7- 19
7- 20
Summary of Position Measures
Measure Definition
Standard score Number of standard deviations that a data value is above or below the
mean
Percentile Position in hundredths that a data value holds in the distribution
Decile Position in tenths that a data value holds in the distribution
Quartile Position in fourths that a data value holds in the distribution
7- 21
Outliers
An outlier is an extremely high or an extremely low data
value when compared with the rest of the data values.
An outlier can strongly affect the mean and standard
deviation of a variable.
This value would then make the mean and standard
deviation of the variable much larger than they really
were.
7- 22
Procedure Table
Procedure for Identifying Outliers
Step1 Arrange the data in order and find Q1 and Q3.
Step2 Find the interquartile range: IQR =Q3 - Q1.
Step3 Multiply the IQR by 1.5.
Step4 Subtract the value obtained in step 3 from Q1 and add the value to Q3
step5 Check the data set for any data value that is smaller than Q1 - 1.5(IQR) or
larger than Q3 +1.5(IQR).
7- 23
Example: Check the following data set for outliers.
5, 6, 12, 13, 15, 18, 22, 50
Solution
The data value 50 is extremely suspect. These are the steps
in checking for an outlier.
Step 1 Find Q1 and Q3.
Q1 is 9 and Q3 is 20.
Step 2 Find the interquartile range (IQR), which is
7- 24
Q3 -Q1. , IQR = Q3 - Q1 =20 -9 = 11
Step 3 Multiply this value by 1.5.
1.5(11) =16.5
Step 4 Subtract the value obtained in step 3 from Q1, and
add the value obtained in step 3 to Q3.
9-16.5= 7.5 and 20 + 16.5 = 36.5
Step 5 Check the data set for any data values that fall
7- 25
outside the interval from 7.5 to 36.5.
The value 50 is outside this interval; hence, it can be
considered an outlier.
7- 26
There are several reasons why outliers may
occur
The data value may have resulted from a measurement
or observational error. Perhaps the researcher measured
the variable incorrectly.
The resulted from a recording error. That is, it may have
been written or typed incorrectly.
The data value may have been obtained from a subject
that is not in the defined population.
7- 27
Boxplot
A boxplot is a graph of a data set obtained by drawing a
horizontal line from the minimum data value to Q1,
drawing a horizontal line from Q3 to the maximum data
value, and drawing a box whose vertical sides pass
through Q1 and Q3 with a vertical line inside the box
passing through the median or Q2.
7- 28
Procedure for constructing a
boxplot
1. Find the five-number summary for the data values,
that is, the maximum and minimum data values, Q1 and
Q3, and the median.
2. Draw a horizontal axis with a scale such that it
includes the maximum and minimum data values.
3. Draw a box whose vertical sides go through Q1 and
Q3, and draw a vertical line though the median.
7- 29
4. Draw a line from the minimum data value to the left
side of the box and a line from the maximum data value to
the right side of the box.
7- 30
The number of meteorites found in 10 states of the United
States is 89, 47, 164, 296, 30, 215, 138, 78, 48, 39.
Construct a boxplot for the data.
Step 1 Arrange the data in order.
Step 2 Find the median Q2=83.5.
Step 3 Find Q1=47.
Step4 Find Q3=164.
Step 5 Draw a scale for the data on the x axis.
47 83.5 164
30 296
The distribution is somewhat positively skewed
7- 31
If the median falls to the left of the center of the box, the
distribution is positively skewed.
If the right line is larger than the left line, the distribution
is positively skewed.
7- 32
Information obtained from a boxplot
If the median falls to the right of the center, the
distribution is negatively skewed. If the left line is larger
than the right line, the distribution is negatively skewed.
If the median is near the center of the box, the
distribution is approximately symmetric.
If the lines are about the same length, the distribution is