# Descriptive Statistics Formulas

## What are Descriptive Statistics?

Descriptive statistics summarize certain aspects of a data set or a population using numeric calculations. Examples of descriptive statistics include:

• mean, average
• midrange
• standard deviation
• quartiles

This calculator generates descriptive statistics for a data set. Enter data values separated by commas or spaces. You can also copy and paste data from spreadsheets or text documents. See allowable data formats in the table below.

## Descriptive Statistics Formulas and Calculations

This calculator uses the formulas and methods below to find the statistical values listed.

### Minimum

Ordering a data set x1 ≤ x2 ≤ x3 ≤ … ≤ xn from lowest to highest value, the minimum is the smallest value x1.

### Maximum

Ordering a data set x1 ≤ x2 ≤ x3 ≤ … ≤ xn from lowest to highest value, the maximum is the largest value xn.

### Range

The range of a data set is the difference between the minimum and maximum.

Range= xnx1

### Sum

The sum is the total of all data values x1 + x2 + x3 + … + xn

Sum=

### Mean

The mean of a data set is the sum of all of the data divided by the size. The mean is also known as the average.

For a Population

μ=

For a Sample

### Median

Ordering a data set x1 ≤ x2 ≤ x3 ≤ … ≤ xn from lowest to highest value, the median is the numeric value separating the upper half of the ordered sample data from the lower half. If n is odd the median is the center value. If n is even the median is the average of the 2 center values.

If n is odd the median is the value at position p where

p=n+12p=n+12
x˜=xpx~=xp

If n is even the median is the average of the values at positions p and p + 1 where

p=n2p=n2
x˜=xp+xp+12x~=xp+xp+12

### Mode

The mode is the value or values that occur most frequently in the data set. A data set can have more than one mode, and it can also have no mode.

### Standard Deviation

Standard deviation is a measure of dispersion of data values from the mean. The formula for standard deviation is the square root of the sum of squared differences from the mean divided by the size of the data set.

For a Population

σ=ni=1(xiμ)2n−−−−−−−−−−−−√σ=∑i=1n(xi−μ)2n

For a Sample

s=ni=1(xix¯¯¯)2n1−−−−−−−−−−−−√s=∑i=1n(xi−x¯)2n−1

### Variance

Variance measures dispersion of data from the mean. The formula for variance is the sum of squared differences from the mean divided by the size of the data set.

For a Population

σ2=ni=1(xiμ)2nσ2=∑i=1n(xi−μ)2n

For a Sample

s2=ni=1(xix¯¯¯)2n1s2=∑i=1n(xi−x¯)2n−1

### Midrange

The midrange of a data set is the average of the minimum and maximum values.

MR=xmin+xmax2MR=xmin+xmax2

### Quartiles

Quartiles separate a data set into four sections. The median is the second quartile Q2. It divides the ordered data set into higher and lower halves.  The first quartile, Q1, is the median of the lower half not including Q2. The third quartile, Q3, is the median of the higher half not including Q2. This is one of several methods for calculating quartiles.

### Interquartile Range

The range from Q1 to Q3 is the interquartile range (IQR).

IQR=Q3Q1IQR=Q3−Q1

### Outliers

Potential outliers are values that lie above the Upper Fence or below the Lower Fence of the sample set.

Upper Fence=Q3+1.5×IQRUpper Fence=Q3+1.5×IQR
Lower Fence=Q11.5×IQRLower Fence=Q1−1.5×IQR

### Sum of Squares

The sum of squares is the sum of the squared differences between data values and the mean.

For a Population

SS=i=1n(xiμ)2SS=∑i=1n(xi−μ)2

For a Sample

SS=i=1n(xix¯¯¯)2