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We will explain the standard normal distribution.
A standard normal distribution is a normal distribution with a mean of 0 and a variance of 1. The standard deviation is the square root of the variance. Therefore, in a standard normal distribution, both the standard deviation and variance are 1.
Let’s take as an example the statistical test scores of 40 nursing college students already mentioned in vol.56. First, the standard values were calculated as shown in Table 1.
[Table 1]
If you create a graph of the frequency distribution and relative frequency of class width 1 for the standard value, Table 2 is obtained.
[Table 2]
When the shape of the relative frequency of the reference value is a normal distribution, this curved distribution is called the standard normal distribution.
In a normal distribution with variable x, mean value m, and standard deviation σ, z=(x-m)÷σ becomes the standard normal distribution, which is also called the z distribution.
Properties of the z distribution (standard normal distribution)
The shape of the z distribution is determined by the mean and standard deviation of the data.
The mean value of the reference value is 0 and the standard deviation is 1; thus, the mean of the z distribution is 0 and the standard deviation is 1.
[Figure 1]
The characteristics of the z distribution are as follows:
・It is symmetrical around the mean value of 0.
・The curve reaches its maximum at the average value and decreases as it spreads leftwards and rightwards.
・If the area surrounded by the curve and the horizontal axis is considered to be 100%, the area of the section within the curve is as shown in Table 3.
[Table 3]
The probability (area) of the interval of the z distribution is calculated using an Excel function.
When calculating the upper probability of horizontal axis x = 1.96:
Lower probability
=NORM.S.DIST(1.96)
= 0.975
Upper probability
=1 -0.975
= 0.025
When finding the horizontal axis x with upper probability p = 0.025:
Horizontal axis x
=NORM.S.INV (lower probability)
=NORM.S.INV(0 .975)
= 1.96
How to check if a frequency distribution is normal
We explained that if the shape of the frequency distribution graph is a symmetrical bell-shaped distribution, then it is a normal distribution; however, if the shape of the peak is too sharp or too flat, it cannot be stated that it is a normal distribution.
Therefore, statistical analysis is required to determine whether the shape of the frequency distribution is a normal distribution.
[Commonly used judgment points to determine whether a distribution is normal]
(1) Judgment based on skewness and kurtosis
(2) Judgment based on normal probability plot
(3) Test of normality
Points (1) and (2) are methods to check whether the frequency distribution created from observed data (sample) is a normal distribution. Point (3) is a method that examines whether the frequency distribution of a population is normal based on the frequency distribution created from data observed through questionnaire surveys and experiments.
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