Quantile-Quantile Plots
Also called
 QQ plots
 Normal probability plots
Uses
 Check whether data came from normal distribution
 Suggest how data is different from normal distribution
 See if two datasets are from the same distribution
General idea
 Using x and s from data, plot sorted data against what you would expect if it were
from a normal distribution.
 Points should follow straight line if normal.
Advantages of QQ-plot over histogram
 Can see if data’s tails are thicker or thinner than a normal distribution.
 Skewness can be more apparent
Disadvantages of QQ-plot
 Not intuitive
 Takes practice
 Styles of QQ-plots differ between software packages, books, etc.
1
Practice Data: n=25, x  108.52, s  19.15
Obtained from Normal   100,   20
data
standardized
values for data
percentiles
Z Quantiles for
percentiles
66
75
83
84
91
94
99
102
103
103
104
107
112
114
115
117
118
118
122
124
125
127
131
133
146
-2.22076
-1.75070
-1.33287
-1.28064
-0.91504
-0.75836
-0.49722
-0.34053
-0.28830
-0.28830
-0.23607
-0.07939
0.18176
0.28621
0.33844
0.44290
0.49513
0.49513
0.70404
0.80850
0.86073
0.96518
1.17410
1.27855
1.95752
0.02
0.06
0.10
0.14
0.18
0.22
0.26
0.30
0.34
0.38
0.42
0.46
0.50
0.54
0.58
0.62
0.66
0.70
0.74
0.78
0.82
0.86
0.90
0.94
0.98
-2.05375
-1.55477
-1.28155
-1.08032
-0.91537
-0.77219
-0.64335
-0.52440
-0.41246
-0.30548
-0.20189
-0.10043
0.00000
0.10043
0.20189
0.30548
0.41246
0.52440
0.64335
0.77219
0.91537
1.08032
1.28155
1.55477
2.05375
Expected data
values using Z
quantiles from
percentiles
69.198
78.751
83.983
87.836
90.994
93.735
96.202
98.480
100.623
102.671
104.654
106.597
108.520
110.443
112.386
114.369
116.417
118.561
120.838
123.305
126.046
129.204
133.057
138.289
147.842
Different software packages use different algorithms for how to divide 0-to-1 up into
equally spaced n points.
Idea: The 10th percentile of the data should match up to what you would expect from a
normal   x,  s  distribution’s 10th percentile, same with 50th percentile, etc.
xi  x
for each
s
data point should match up with the p th percentile for the standard normal distribution.
Equivalent idea: The p th percentile for the standardized value xi, standardized 
2
Probability Plot of data
Normal
99
95
90
Mean
StDev
N
AD
P-Value
108.5
19.15
25
0.214
0.831
Mean
StDev
N
AD
P-Value
108.5
19.15
25
0.214
0.831
Percent
80
70
60
50
40
30
20
10
5
1
60
70
80
90
100
110
data
120
130
140
150
Probability Plot of data
Normal - 95% CI
99
95
90
Percent
80
70
60
50
40
30
20
10
5
1
50
75
100
125
150
175
data
The first graph is from Stat>Basic Statistics>Normality test. The second graph is from
from Graph > Probability Plot. The 2nd graph includes a 95% pointwise confidence
interval for what you would expect if the data came from a normal distribution. The pvalues are from the Anderson Darling test for normality. There are a number of tests for
normality.
3
120
80
Sample Quantiles
Normal Q-Q Plot
-2
-1
0
1
2
Theoretical Quantiles
2
1
0
-1
-2
Theoretical Quantiles
Normal Q-Q Plot
80
100
120
140
2
1
0
-1
-2
Standard Normal Quantiles
Sample Quantiles
-2
-1
0
1
2
127
146
Percents
Data Z-scores
0.98
0.94
0.78
0.62
0.5
0.38
0.18
0.06
0.02
66
75
83
91
99
107
117
Data values
4
Note how Minitab Probability Plot’s y-axis has unequal spacing for the percents
Some standard normal quantiles
1.96
1.282
0.842
0.524
0.253
0
-0.253
-0.524
-0.842
-1.282
-1.96
percents
0.975
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.025
quanitles
The above 4 plots were programmed in R.
Probability Plot of data
Normal
99
Mean
StDev
N
AD
P-Value
95
90
108.5
19.15
25
0.214
0.831
Percent
80
70
60
50
40
30
20
10
5
1
60
70
80
90
100
110
data
120
130
140
150
5
Histogram of Chi-sq df=4
Normal
90
Mean
StDev
N
80
4.011
3.009
500
70
Frequency
60
50
40
30
20
10
0
-3
0
3
6
9
12
Chi-sq df=4
15
18
Probability Plot of Chi-sq df=4
Normal - 95% CI
99.9
Mean
StDev
N
AD
P-Value
99
95
Percent
90
4.011
3.009
500
13.472
<0.005
80
70
60
50
40
30
20
10
5
1
0.1
-5
0
5
10
Chi-sq df=4
15
20
Histogram of Left Skewed
Normal
90
Mean
StDev
N
80
35.99
3.009
500
70
Frequency
60
50
40
30
20
10
0
21
24
27
30
33
Left Skewed
36
39
42
Probability Plot of Left Skewed
Normal - 95% CI
99.9
Mean
StDev
N
AD
P-Value
99
95
Percent
90
35.99
3.009
500
13.472
<0.005
80
70
60
50
40
30
20
10
5
1
0.1
20
25
30
35
Left Skewed
40
45
6
Histogram of t-distn, df=4
Normal
200
Mean
StDev
N
0.06165
1.356
500
Mean
StDev
N
AD
P-Value
0.06165
1.356
500
4.404
<0.005
Mean
StDev
N
-0.1607
1.399
20
Frequency
150
100
50
0
-3
0
3
6
t-distn, df=4
9
12
Probability Plot of t-distn, df=4
Normal - 95% CI
99.9
99
95
90
Percent
80
70
60
50
40
30
20
10
5
1
0.1
-5
0
5
t-distn, df=4
10
15
Histogram of t, df=4, n=20
Normal
9
8
Frequency
7
6
5
4
3
2
1
0
-4
-3
-2
-1
0
t, df=4, n=20
1
2
3
Probability Plot of t, df=4, n=20
Normal - 95% CI
99
Mean
StDev
N
AD
P-Value
95
90
-0.1607
1.399
20
0.733
0.047
Percent
80
70
60
50
40
30
20
10
5
1
-5.0
-2.5
0.0
t, df=4, n=20
2.5
5.0
7
Histogram of Beta(2,2)
Normal
40
Mean
StDev
N
0.5039
0.2308
500
Mean
StDev
N
AD
P-Value
0.5039
0.2308
500
2.741
<0.005
Frequency
30
20
10
0
-0.00
0.15
0.30
0.45
0.60
Beta(2,2)
0.75
0.90
1.05
Probability Plot of Beta(2,2)
Normal - 95% CI
99.9
99
95
90
Percent
80
70
60
50
40
30
20
10
5
1
0.1
-0.4
-0.2
0.0
0.2
0.4
0.6
Beta(2,2)
0.8
1.0
1.2
1.4
Can be used to plot random values against random values to if distributions are the same.
0
-1
-2
Znorm
1
2
30 random values from two distributions
-2
-1
0
1
2
tDistnDF4
8