Assessing normality

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BIOL 4243 IUG

59

Assessing normality

Not all continuous random variables are normally distributed. It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution. In this section some statistical tools will be presented to check whether a given set of data is normally distributed.

1. Previous knowledge of the nature of the distribution

Problem: A researcher working with sea stars needs to know if sea star size (length of radii) is normally distributed. What do we know about the size distributions of sea star populations?

1. Has previous work with this species of sea star shown them to be normally

distributed?

2. Has previous work with a closely related species of seas star shown them to be

normally distributed?

3. Has previous work with seas stars in general shown them to be normally

distributed? If you can answer yes to any of the above questions and you do not have a reason to think your population should be different, you could reasonably assume that your population is also normally distributed and stop here. However, if any previous work has shown non-normal distribution of sea stars you had probably better use other techniques.

2. Construct charts

For small- or moderate-sized data sets, the stem-and-leaf display and box-and- whisker plot will look symmetric. For large data sets, construct a histogram or polygon and see if the distribution bell-shaped or deviates grossly from a bell-shaped normal distribution. Look for skewness and asymmetry. Look for gaps in the distribution - intervals with no observations. However, remember that normality requires more than just symmetry; the fact that the histogram is symmetric does not mean that the data come from a normal distribution. Also, data sampled from normal distribution will sometimes look distinctly different from the parent distribution. So, we need to develop some techniques that allow us to determine if data are significantly different from a normal distribution.

3. Normal Counts method

Count the number of observations within 1, 2, and 3 standard deviations of the mean and compare the results with what is expected for a normal distribution in the 68-95-

99.7 rule. According to the rule,

68% of the observations lie within one standard deviation of the mean.

95% of observations within two standard deviations of the mean.

99.7% of observations within three standard deviations of the mean.

Example: As part of a demonstration one semester, I collected data on the heights of sample of 25 IUG biostatistics students. These data are presented in the table below. Does the sample shown below have been drawn from normally distributed populations?

BIOL 4243 IUG

60
Table. Heights, in inches, of 25 IUG biostatistics students.

71.0 69.0 70.0 72.5 73.0

70.0 71.5 70.5 72.0 71.0

68.5 69.0 69.0 68.5 74.0

67.0 69.0 71.5 66.0 70.0

68.5 74.0 74.5 74.0

Solution:

For normal Counts method, determine the following

Heights, in

inches

Frequency

66 1
67 1
68.5
3 69 4
70 3

70.5 1

71 2

Total = 17

71.5 2

72 1

72.5 1

73 1
74 3

74.5 1

Total 24

x = 70.6; s = 2.3 xs is 72.9 to 68.3.

17 out of the 24 observations i.e. 17/24 = 0.70 = 70% fall within

xs, i.e. between

72.9 and 68.3, which is approximately equal to 68%.There is no reason to doubt that

the sample is drawn from a normal population.

4. Compute descriptive summary measures

a. The mean, median and mode will have similar values. b. The interquartile range approximately equal to 1.33 s. c. The range approximately equal 6 s.

5. Evaluate normal probability plot

If the data come from a normal or approximately normal distribution, the plotted points will fall approximately along a straight line (a 45 degree line). However, if your sample departs from normality, the points on the graph will deviate from that line.

BIOL 4243 IUG

59

Assessing normality

Not all continuous random variables are normally distributed. It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution. In this section some statistical tools will be presented to check whether a given set of data is normally distributed.

1. Previous knowledge of the nature of the distribution

Problem: A researcher working with sea stars needs to know if sea star size (length of radii) is normally distributed. What do we know about the size distributions of sea star populations?

1. Has previous work with this species of sea star shown them to be normally

distributed?

2. Has previous work with a closely related species of seas star shown them to be

normally distributed?

3. Has previous work with seas stars in general shown them to be normally

distributed? If you can answer yes to any of the above questions and you do not have a reason to think your population should be different, you could reasonably assume that your population is also normally distributed and stop here. However, if any previous work has shown non-normal distribution of sea stars you had probably better use other techniques.

2. Construct charts

For small- or moderate-sized data sets, the stem-and-leaf display and box-and- whisker plot will look symmetric. For large data sets, construct a histogram or polygon and see if the distribution bell-shaped or deviates grossly from a bell-shaped normal distribution. Look for skewness and asymmetry. Look for gaps in the distribution - intervals with no observations. However, remember that normality requires more than just symmetry; the fact that the histogram is symmetric does not mean that the data come from a normal distribution. Also, data sampled from normal distribution will sometimes look distinctly different from the parent distribution. So, we need to develop some techniques that allow us to determine if data are significantly different from a normal distribution.

3. Normal Counts method

Count the number of observations within 1, 2, and 3 standard deviations of the mean and compare the results with what is expected for a normal distribution in the 68-95-

99.7 rule. According to the rule,

68% of the observations lie within one standard deviation of the mean.

95% of observations within two standard deviations of the mean.

99.7% of observations within three standard deviations of the mean.

Example: As part of a demonstration one semester, I collected data on the heights of sample of 25 IUG biostatistics students. These data are presented in the table below. Does the sample shown below have been drawn from normally distributed populations?

BIOL 4243 IUG

60
Table. Heights, in inches, of 25 IUG biostatistics students.

71.0 69.0 70.0 72.5 73.0

70.0 71.5 70.5 72.0 71.0

68.5 69.0 69.0 68.5 74.0

67.0 69.0 71.5 66.0 70.0

68.5 74.0 74.5 74.0

Solution:

For normal Counts method, determine the following

Heights, in

inches

Frequency

66 1
67 1
68.5
3 69 4
70 3

70.5 1

71 2

Total = 17

71.5 2

72 1

72.5 1

73 1
74 3

74.5 1

Total 24

x = 70.6; s = 2.3 xs is 72.9 to 68.3.

17 out of the 24 observations i.e. 17/24 = 0.70 = 70% fall within

xs, i.e. between

72.9 and 68.3, which is approximately equal to 68%.There is no reason to doubt that

the sample is drawn from a normal population.

4. Compute descriptive summary measures

a. The mean, median and mode will have similar values. b. The interquartile range approximately equal to 1.33 s. c. The range approximately equal 6 s.

5. Evaluate normal probability plot

If the data come from a normal or approximately normal distribution, the plotted points will fall approximately along a straight line (a 45 degree line). However, if your sample departs from normality, the points on the graph will deviate from that line.