Methods and formulas for Descriptive Statistics

Select the method or formula of your choice.

Coefficient of variation (CoefVar)

The coefficient of variation is a measure of relative variability calculated as a percentage.

Formula

Minitab calculates it as:

Notation

TermDescription
s standard deviation of the sample
mean of the observations

Interquartile range (IQR)

The interquartile range equals the third quartile minus the 1st quartile.

Kurtosis

Kurtosis is one measure of how different a distribution is from the normal distribution. A positive value usually indicates that the distribution has a sharper peak than the normal distribution. A negative value indicates that the distribution has a flatter peak than the normal distribution.

Formula

Notation

TermDescription
xi i th observation
mean of the observations
N number of nonmissing observations
s standard deviation of the sample

Maximum

The largest value in your data set.

Mean

A commonly used measure of the center of a batch of numbers. The mean is also called the average. It is the sum of all observations divided by the number of (nonmissing) observations.

Formula

Notation

TermDescription
xiith observation
Nnumber of nonmissing observations

Median

The sample median is in the middle of the data: at least half the observations are less than or equal to it, and at least half are greater than or equal to it.

Suppose you have a column that contains N values. To calculate the median, first order your data values from smallest to largest. If N is odd, the sample median is the value in the middle. If N is even, the sample median is the average of the two middle values.

For example, when N = 5 and you have data x1, x2, x3, x4, and x5, the median = x3.

When N = 6 and you have ordered data x1, x2, x3, x4, x5,and x6:

where x3 and x4 are the third and fourth observations.

Minimum

The smallest value in your data set.

Mode

The mode is the data value that occurs most often in the dataset. If multiple modes exist, Minitab displays the smallest modes, up to a total of four. N for Mode is the number of times the mode (or modes) appears.

N

The number of non-missing values in the sample.

In this example, there are 141 recorded observations.
Total count N N*
149 141 8

N*

The number of missing values in the sample. The number of missing values refers to cells that contain the missing value symbol *.

In this example, 8 errors occurred during data collection and are recorded as missing values.
Total count N N*
149 141 8

Total Count

The total number of observations in the column. Use to represent the sum of N missing and N nonmissing.

In this example, there are 141 valid observations and 8 missing values. The total count is 149.
Total count N N*
149 141 8

Range

The range is calculated as the difference between the largest and smallest data value.

R = Maximum – Minimum

Skewness

Skewness is a measure of asymmetry. A negative value indicates skewness to the left, and a positive value indicates skewness to the right. A zero value does not necessarily indicate symmetry.

Formula

Notation

TermDescription
xi i th observation
mean of the observations
N number of nonmissing observations
s standard deviation of the sample

Standard deviation (StDev)

The sample standard deviation provides a measure of the spread of your data. It is equal to the square root of the sample variance.

Formula

If the column contains x 1, x 2,..., x N, with mean , then the standard deviation of the sample is:

Notation

TermDescription
x i i th observation
mean of the observations
N number of nonmissing observations

Standard error of the mean (SE Mean)

The standard error of the mean is calculated as the standard deviation divided by the square root of the sample size.

Formula

Notation

TermDescription
s standard deviation of the sample
N number of nonmissing observations

Sum

Formula

Notation

TermDescription
xi i th observation

1st quartile (Q1)

25% of your sample observations are less than or equal to the value of the 1st quartile. Therefore, the 1st quartile is also referred to as the 25th percentile.

Formula

Notation

TermDescription
ytruncated integer value of w
w
zfraction component of w that was truncated
xjjth observation in the list of sample data, ordered from smallest to largest
Note

When w is an integer, y = w, z = 0, and Q1 = xy.

3rd quartile (Q3)

75% of your sample observations are less than or equal to the value of the third quartile. Therefore, the third quartile is also referred to as the 75th percentile.

Formula

Notation

TermDescription
ytruncated value of w
w
zfraction component of w that was truncated away
xjjth observation in the list of sample data, ordered from smallest to largest
Note

When w is an integer, y = w, z = 0, and Q3 = xy.

Variance

The variance measures how spread out the data are about their mean. The variance is equal to the standard deviation squared.

Formula

Notation

TermDescription
xiith observation
mean of the observations
Nnumber of nonmissing observations
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