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)
