This document discusses various measures of position such as percentiles, quartiles, and standard scores. It provides examples of how to calculate percentiles, quartiles, z-scores, and identify outliers. Key points covered include how to interpret percentiles and z-scores, how to calculate quartiles and find values corresponding to specific percentiles, and the procedure for identifying outliers in a data set.

1. S E C T I O N 3 - 4
Measures of Position

2. Objective
 To identify the position of a data value in a
data set, using various measures of position,
such as percentiles, deciles, and quartiles.

3. Percentile Example
 If a data value is located at the 80th percentile,
it means that 80% of the values fall below it in
the distribution and 20% fall above it.

4. Standard Scores
 How can I compare a score of 32 on my social
studies test to a score of 115 on my math test?
 Z-Scores!
 Allow us to make a comparison of unrelated raw
scores
 A Z-Score (Standard Score) for a value is
obtained by subtracting the mean from the
value and dividing the result by the standard
deviation.

5. Z-Score
 Symbol: z
 Formula: z = value – mean
standard deviation
 Samples:
 Populations:
X X
z
s


X
z





6. Z-Score Example – p.145 #14
 A student scores 60 on a mathematics test that
has a mean of 54 and a standard deviation of 3,
and she scores 80 on a history test with a mean
of 75 and a standard deviation of 2. On which
test did she perform better?

7. Answer:
 Math test z-score: 2.0
 History test z-score: 2.5
 She performed better on the history test!

8. Important Point
 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.
 A z-score, then, is actually the number of
standard deviations each value is from the
mean for a specific distribution.

9. Percentiles
 Percentiles divide the data set in 100 equal
groups.
 See p. 136 for an example (weights of girls by
age and percentile rankings)
 Find the percentile ranking for an 18-year-old
girl who weighs 100 lb.
 What does the 5th percentile mean here?
 Means: 5 percent of 18-year-old girls weigh
100 lb or less. (at or below 100 lb)

10. Constructing a Percentile Graph, p. 145 #20
 The airborne speeds in miles per hour of 21
planes are shown. Find the approximate values
that correspond to the given percentiles by
constructing a percentile graph.
Class Frequency
366-386 4
387-407 2
408-428 3
429-449 2
450-470 1
471-491 2
492-512 3
513-533 4
Σf = 21

11. Constructing a Percentile Graph, p. 145 #20
 The airborne speeds in miles per hour of 21
planes are shown. Find the approximate values
that correspond to the given percentiles by
constructing a percentile graph.
Class Frequency Cumulative
Frequency
Cumulative
Percent
366-386 4 4 4/21= 19%
387-407 2 6 6/21 = 21%
408-428 3 9 9/21 = 43%
429-449 2 11 11/21 = 52%
450-470 1 12 12/21 = 57%
471-491 2 14 14/21 = 67%
492-512 3 17 17/21 = 81%
513-533 4 21 21/21 = 100%
Σf = 21

12. Percentile Graph for p. 145 #20
0
10
20
30
40
50
60
70
80
90
100
366 386 407 428 449 470 491 512 533
Cumulative%
Class Boundaries – Airborne Speed
a) 45th percentile
b) 9th percentile
c) 20th percentile
d) 60th percentile
e) 75th percentile

13. Percentile Formula
 The percentile corresponding to a given value X is
computed by using the following formula:
 Percentile = (number of values below X) + 0.5 * 100
total number of values

14. Percentile Example
 p. 145 # 22
 Find the percentile ranks of each weight in the
data set. The weights are in pounds.
 Data Set: 78, 82, 86, 88, 92, 97
 78: (0 + .5)/6 = 0.083333…* 100 = 8th percentile
 82: (1 + .5)/6 = .25 * 100 = 25th percentile
 86: (2 + .5)/6 = .416666…*100 = 42nd percentile
 88: (3 + .5)/6 = .583333…*100 = 58th percentile
 92: (4 + .5)/6 = .75 * 100 = 75th percentile
 97: (5 + .5)/6=.91666…*100 = 92nd percentile

15. Procedure for Finding a Data Value Corresponding to a
Given Percentile
 Arrange the data in order from lowest to highest.
 Substitute into the formula c = n * p
100
where n = total number of values and p = percentile
 If c is not a whole number, round up to the next
whole number. Starting at the lowest value, count
over to the number that corresponds to the
rounded-up value.
 If c is a whole number, use the value halfway
between the cth and (c+1)st values when counting
up from the lowest value.

16. Percentile Rank Example, p. 145 # 23
 What value corresponds to the 30th percentile?
 Data Set: 78, 82, 86, 88, 92, 97
 Substitute into the formula c = n * p
100
where n = total number of values and p = percentile
 C = (6 * 30)/100 = 1.8 (round up to 2 or 2nd
number)
 30th percentile for this set would be 82.

17. Quartiles and Deciles
 Quartiles divide the distribution into four
groups, separated by Q1, Q2, and Q3.
 Q1 is the same as the 25th percentile.
 Q2 is the same as the 50th percentile or the
median.
 Q3 is the same as the 75th percentile.

18. Procedure for Finding Data Values Corresponding to Q1,
Q2, and Q3
 Arrange the data in order from lowest to
highest.
 Find the median of the data values. This is the
value for Q2.
 Find the median of the data values that fall
below Q2. This is the value for Q1.
 Find the median of the data values that fall
above Q2. This is the value for Q3.

19. Example 3-36, p. 141
 Find Q1, Q2, and Q3 for the data set: 15, 13, 6,
5, 12, 50, 22, 18.
 Arrange in order: 5, 6, 12, 13, 15, 18, 22, 50
 Find the median or Q2. This is an even set of
data, so find the two in the middle and find
their midpoint. (13+15)/2 = 28 /2 = 14.
 Find Q1. This is the median of the numbers less
than 14, or 5, 6, 12, and 13. (6+12)/2 = 18/2
= 9.
 Find Q3. Median of 15, 18, 22, and 50.
(18+22)/2 = 40/2 = 20.

20. Interquartile Range (IQR)
 The difference between Q1 and Q3 (Q3 - Q1).
 Used to identify outliers.
 Used as a measure of variability in exploratory
analysis.

21. Deciles
 Divide a distribution into 10 groups. Denoted
D1, D2, D3, etc.
 Correspond to P10, P20, P30, etc.

22. Outliers
 Outliers are extremely high or low values.
 Can strongly affect the mean and standard
deviation of a variable.

23. Procedure for Identifying Outliers
1. Arrange the data in order and find Q1 and Q3.
2. Find the IQR (Q3 – Q1).
3. Multiply the IQR by 1.5.
4. Subtract the value obtained in step 3 from Q1
and add the value to Q3.
5. Check the data set for any data value that is
smaller than Q1 - 1.5(IQR) or larger than
Q3+1.5(IQR)

24. Outlier Example, p. 146 #30a
 Check the following set for outliers:
 16, 18, 22, 19, 3, 21, 17, 20
 3, 16, 17, 18, 19, 20, 21, 22
 Q2 = 18.5, Q1 = 16.5, Q3 = 20.5
 IQR = 20.5 – 16.5 = 4
 4*1.5 = 6
 Upper boundary: 20.5 + 6 = 26. 5 **No upper
outliers
 Lower boundary: 16.5 – 6 = 10.5 **Lower
outliers: 3

25. Assignment:
 p.144-146, #1-9, 11-15 odd, 24, 30 b,d,e, and
31