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Normality of the distribution

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Some statistical tools assume that the distribution is normal. The algorithm for checking the normality of the distribution will be given below, and also an example in excel.

Distribution law

Checking for compliance with the normal distribution is a special case of solving the problem on finding among the known distribution functions one that describes as accurately as possible this distribution.

First of all, it is necessary to structure the available values, in the article properties distributions it describes how the distribution series is constructed, so here I will omit the details and give source data and processed values:

146 136 169 147 157 160 153 148 149 133
165 156 127 162 157 135 136 147 149 140
167 139 139 128 139 131 157 161 164 154
154 161 156 162 157 145 146 135 163 162
142 154 161 149 162 147 135 139 152 150
171 146 144 143 146 158 147 160 163 141
155 153 153 156 166 168 131 134 157 143
147 145 147 164 172 136 150 156 145 149
142 157 158 164 141 161 143 166 146 153
130 135 168 141 159 149 143 142 139 154
Table 1. Initial data for checking the normality of the distribution
# 12345678910
x56911207171194
pi0.050.060.090.110.20.070.170.110.090.04
Table 2. Number of elements in each interval
Graph 1. Distribution range

Regardless of what we see on the graph, we need to check whether whether the distribution is normal.

The characteristics of a normal distribution are the mean and standard deviation. Let's calculate these values for our distribution:

μ = 150.2
σ = 10.81
The calculation of the mean and standard deviation is described in the article distribution parameters

Normal distribution

The normal distribution curve for μ=150.2 and σ=
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:

P(x) = e^[-0.5((x-150.2)/10.81)2] / [10.81√2π] Normal distribution formula
Graph 2. Distribution series and normal distribution, μ = 150.2, σ = 10.81

First approximation

Let's try to invent a criterion of normality, the simplest, what comes to mind is to determine the percentage of compliance the normal curve and the existing distribution.

To do this, add up the absolute values of the differences across all points of the graph, find the area under the normal distribution graph and calculate the deviation of interest, I will call such a criterion "criterion of normality" and I will decide that if the deviation more, let's say 30%, then the distribution is not normal.

diff = Σ|D(X) - P(X)|
S = ΣP(X)
Δ = diff / S
diff = 58.37
S = 41.63
Δ = 140%

The deviation is 140%, therefore, I draw the following conclusion: the distribution is not normal according to the normality criterion.

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Last edit time: 04.09.2025 k-tree

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