Dolly Robinson

Answered

2020-11-17

To calculate normal distributions values must be?

Answer & Explanation

lamusesamuset

Expert

2020-11-18Added 93 answers

Step 1

Standard normal distribution:

The standard normal distribution is a special case of normal distribution, in which the mean of the distribution is 0 and standard deviation of the distribution is 1. The Z-scores for the given sample size can be calculated using the given formula.

$z=\frac{x-\mu}{\sigma}/\left(\sqrt{n}\right)$

where, x is the normal random variable, n is the sample size,$\mu$ is the mean, and $\sigma$ is the standard deviation

Step 2

The procedure for obtaining the percentage of all the possible observations that lie within the specified range is as follows:

-Sketch the normal curve associated with the variable.

-Shade the region of interest and mark the delimiting x-values.

-Compute the z-scores for the x-

-Find the area under the standard normal curve for the computed z-scores using standard normal table.

Several tests of normality exist, using which you can verify whether a particular data follows the normal distribution.

Usually, before conducting a formal test, we prefer to take the help of graphical methods, to see if the data may be assumed to follow the normal distribution, at least approximately. A few such graphical methods are:

-Histogram of the data , superimposed with a normal probability curve,

-Normal probability plot with confidence interval,

-Normal quantile-quantile (QQ) plot.

-Boxplot, etc.

Step 3

If the graphical display appears to show at least an approximate normal distribution, then a formal test can be used to verify the normality. A few such tests are as follows:

-Pearson’s Chi-squared test for goodness of fit,

-Shapiro-Wilk test,

-Kolmogorov-Smirnov test, etc.

The Pearson’s Chi-squared test is discussed here.

Pearson’s Chi-squared test for goodness of fit:

Suppose the data set can be divided into n categories or classes, with observed frequency in the$i}^{th$ class as $O}_{i$ and expected frequency in the $i}^{th$ class as $E}_{i$ (i = 1, 2, …, n). Further, assume that the data is obtained from a simple random sampling, the total sample size is large, each cell count (for each category) is at least 5 and the observations are independent.

Then, the degrees of freedom, df = (number of categories) – (number of parameters in the model) – 1. For n categories in the data set and 2 parameters (mean and variance) of the normal distribution, df = n – 3.

The test statistic for the test is given as,${\chi}^{2}=\sum \left[\frac{{({O}_{i}-{E}_{i})}^{2}}{{E}_{i}}\right]$ ,where the summation is done over all i = 1, 2, …, n.

The observed frequencies will be known from the data set. The expected frequencies for a normal distribution can be obtained by multiplying the total sample size, say, N, by the normal probability for the corresponding class (obtained from a standard normal table or any software such as, EXCEL, MINITAB, etc.).

The corresponding p-value for the test can be used to check whether the data follows normal distribution or not.

Assumptions of normality:

The assumptions of normality are as follows:

-The data should be symmetric.

-The data should be mesokurtic.

-The empirical rule must be satisfied.

Empirical rule:

-68% of values fall within one standard deviation from the mean.

-95% of values fall within two standard deviations from the mean.

-7% of values fall within three standard deviations from the mean.

Standard normal distribution:

The standard normal distribution is a special case of normal distribution, in which the mean of the distribution is 0 and standard deviation of the distribution is 1. The Z-scores for the given sample size can be calculated using the given formula.

where, x is the normal random variable, n is the sample size,

Step 2

The procedure for obtaining the percentage of all the possible observations that lie within the specified range is as follows:

-Sketch the normal curve associated with the variable.

-Shade the region of interest and mark the delimiting x-values.

-Compute the z-scores for the x-

-Find the area under the standard normal curve for the computed z-scores using standard normal table.

Several tests of normality exist, using which you can verify whether a particular data follows the normal distribution.

Usually, before conducting a formal test, we prefer to take the help of graphical methods, to see if the data may be assumed to follow the normal distribution, at least approximately. A few such graphical methods are:

-Histogram of the data , superimposed with a normal probability curve,

-Normal probability plot with confidence interval,

-Normal quantile-quantile (QQ) plot.

-Boxplot, etc.

Step 3

If the graphical display appears to show at least an approximate normal distribution, then a formal test can be used to verify the normality. A few such tests are as follows:

-Pearson’s Chi-squared test for goodness of fit,

-Shapiro-Wilk test,

-Kolmogorov-Smirnov test, etc.

The Pearson’s Chi-squared test is discussed here.

Pearson’s Chi-squared test for goodness of fit:

Suppose the data set can be divided into n categories or classes, with observed frequency in the

Then, the degrees of freedom, df = (number of categories) – (number of parameters in the model) – 1. For n categories in the data set and 2 parameters (mean and variance) of the normal distribution, df = n – 3.

The test statistic for the test is given as,

The observed frequencies will be known from the data set. The expected frequencies for a normal distribution can be obtained by multiplying the total sample size, say, N, by the normal probability for the corresponding class (obtained from a standard normal table or any software such as, EXCEL, MINITAB, etc.).

The corresponding p-value for the test can be used to check whether the data follows normal distribution or not.

Assumptions of normality:

The assumptions of normality are as follows:

-The data should be symmetric.

-The data should be mesokurtic.

-The empirical rule must be satisfied.

Empirical rule:

-68% of values fall within one standard deviation from the mean.

-95% of values fall within two standard deviations from the mean.

-7% of values fall within three standard deviations from the mean.

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