These notes are excerpted from the text Quantitative Methods, by Esch, Bukowski, and Kruse. You
might find them useful in helping to determine if a given data set is normally distributed.
Normal Curve
Although frequency distributions can be of any shape (unimodal, bimodal, peaked, flat,
skewed, etc.), there is one pattern that occurs in such a wide variety of circumstances that it
is regarded as the single most important type of data distribution in all of statistics. Its
proper name is the normal distribution (or Gaussian distribution), but we often hear it
referred to as the “bell-shaped” distribution, or the “bell curve”. It is unimodal, perfectly
symmetrical and bell-shaped, with tails extending indefinitely in both directions approaching
but never quite touching the horizontal axis.
Normal distributions occur for data ranging from heights and weights and IQ’s of people to
distances and volumes and speeds of heavenly bodies. They also occur commonly for such
populations as lifetimes of batteries, actual lengths of 1 nails, magnitudes of errors of
electronic scanners, family incomes, incubation times for baby chicks, and so on. In fact the
normal distribution occurs so commonly and has so many applications that it is sometimes
called the “cornerstone of modern statistics”.
What does “bell-shaped” mean exactly? There are lots of shapes that could be described as
bell-type shapes, such as below:
However, we want a very special bell, one that looks like the bell in the normal curve – and
we want our histograms to resemble this bell. In fact, if our data are normally distributed,
then ALL of their frequency distributions (represented by bar charts and histograms) will
resemble this special bell. Of course, the smaller the intervals and the greater the number of
bins, the more the histogram will resemble a bell.
It turns out, that if data are perfectly normally distributed, then the pattern is so precise that it
can be totally described by knowing just the mean and the standard deviation. Given ANY
number z of standard deviations it is possible to determine EXACTLY what percent of the data
should be within that number of standard deviations from the mean, i.e., between z.
And once we know this, we know exactly what our histograms should look like – and
exactly what “bell-shaped” means.
With normally distributed data, we know that (rounded to the nearest percent) PRECISELY
68% of the data fall within one standard deviation of the mean, and PRECISELY 95% of the
data fall within two standard deviations, etc. In fact, for any number (or fraction) of standard
deviations, we know EXACTLY what percent of the data fall within that number of standard
deviations of the mean.
For example, consider the normal curve percentages in the table below. Given a number z of
standard deviations, the table gives the percent of data values between the mean  and z.
Note that since this only includes data values above the mean, this is only half of the percent
of data values within z standard deviations of the mean, i.e., between z. Also, the
percents in the table are cumulative; they all start at . For example, (rounded to two
decimal places) 19.15% of the values will fall between  and +0.5, 34.13% fall between 
and +1, 43.32% fall between  and +1.5, etc. The percentages in the graph however
are not cumulative. They are percents between two +z values. For example, for normally
distributed data, (rounded to two decimal places) 19.15% of the values will fall between  =
+0 and +0.5, 14.98% between +0.5 and +1, 9.19% between +1 and +1.5,
etc.
z
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Normal Curve
Percentages
for [,z]
(Cumulative)
0.00
19.15
34.13
43.32
47.72
49.38
49.87
49.98
49.99
19.15
Normal Curve
Percentages
14.98
(Not Cumulative)
9.19
4.4
2.15
0.13
z
   

z
  

The above Normal Curve Percentages table lists only a few (9) of the possible z values. The
Normal Curve Percentages table below lists many more (330).
Since the heights of the columns (bars) in a bar chart represent numbers (frequencies) of data
values, and the widths of the bars are uniform, the areas of the bars therefore represent
proportions (percentages) of the total population of data values that fall within certain
ranges. Similarly the area under the normal curve can be interpreted in terms of percentages
of the total population falling within certain ranges of values.
Normal Curve Percentages
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
+0.00
0.00
3.98
7.93
11.79
15.54
19.15
22.57
25.80
28.81
31.59
34.13
36.43
38.49
40.32
41.92
43.32
44.52
45.54
46.41
47.13
47.72
48.21
48.61
48.93
49.18
49.38
49.53
49.65
49.74
49.81
49.87
+0.01
0.40
4.38
8.32
12.17
15.91
19.50
22.91
26.11
29.10
31.86
34.38
36.65
38.69
40.49
42.07
43.45
44.63
45.64
46.49
47.19
47.78
48.26
48.64
48.96
49.20
49.40
49.55
49.66
49.75
49.82
49.87
+0.02
0.80
4.78
8.71
12.55
16.28
19.85
23.24
26.42
29.39
32.12
34.61
36.86
38.88
40.66
42.22
43.57
44.74
45.73
46.56
47.26
47.83
48.30
48.68
48.98
49.22
49.41
49.56
49.67
49.76
49.82
49.87
+0.03
1.20
5.17
9.10
12.93
16.64
20.19
23.57
26.73
29.67
32.38
34.85
37.08
39.07
40.82
42.36
43.70
44.84
45.82
46.64
47.32
47.88
48.34
48.71
49.01
49.25
49.43
49.57
49.68
49.77
49.83
49.88
+0.04
1.60
5.57
9.48
13.31
17.00
20.54
23.89
27.04
29.95
32.64
35.08
37.29
39.25
40.99
42.51
43.82
44.95
45.91
46.71
47.38
47.93
48.38
48.75
49.04
49.27
49.45
49.59
49.69
49.77
49.84
49.88
+0.05
1.99
5.96
9.87
13.68
17.36
20.88
24.22
27.34
30.23
32.89
35.31
37.49
39.44
41.15
42.65
43.94
45.05
45.99
46.78
47.44
47.98
48.42
48.78
49.06
49.29
49.46
49.60
49.70
49.78
49.84
49.89
+0.06
2.39
6.36
10.26
14.06
17.72
21.23
24.54
27.64
30.51
33.15
35.54
37.70
39.62
41.31
42.79
44.06
45.15
46.08
46.86
47.50
48.03
48.46
48.81
49.09
49.31
49.48
49.61
49.71
49.79
49.85
49.89
+0.07
2.79
6.75
10.64
14.43
18.08
21.57
24.86
27.94
30.78
33.40
35.77
37.90
39.80
41.47
42.92
44.18
45.25
46.16
46.93
47.56
48.08
48.50
48.84
49.11
49.32
49.49
49.62
49.72
49.79
49.85
49.89
+0.08
3.19
7.14
11.03
14.80
18.44
21.90
25.17
28.23
31.06
33.65
35.99
38.10
39.97
41.62
43.06
44.29
45.35
46.25
46.99
47.61
48.12
48.54
48.87
49.13
49.34
49.51
49.63
49.73
49.80
49.86
49.90
+0.09
3.59
7.53
11.41
15.17
18.79
22.24
25.49
28.52
31.33
33.89
36.21
38.30
40.15
41.77
43.19
44.41
45.45
46.33
47.06
47.67
48.17
48.57
48.90
49.16
49.36
49.52
49.64
49.74
49.81
49.86
49.90
3.5
49.977
49.978
49.978
49.979
49.980
49.981
49.982
49.982
49.983
49.984
4.0
49.997
49.997
49.997
49.997
49.997
49.997
49.998
49.998
49.998
49.998