Statistics Calculator

Find the descriptive summary statistics for a set of data by entering the numbers below. Keep reading to learn how to calculate each one.

Separate numbers using a comma (,)
Separate numbers using a comma (,)

Have a Question or Feedback?

Results:

Min:
 
Max:
 
Count:
 
Range:
 
Sum:
 
Mean:
 
Median:
 
Mode:
 
Standard Deviation:
 
Variance:
 
Sum of Squares:
 
Quartile Q1:
 
Quartile Q2:
 
Quartile Q3:
 
Interquartile Range:
 
Midrange:
 
Mean Absolute Deviation:
 
Coefficient of Variation:
 
Relative Standard Deviation:
 
Learn how we calculated this below

On this page:

By
Bio image of Joe Sexton

Joe is the creator of Inch Calculator and has over 20 years of experience in engineering and construction. He holds several degrees and certifications.

Full bio

Review Checkmark
Reviewed by
Bio image of Brian Libgober, PhD

Brian specializes in political science and statistics with several advanced degrees from Harvard.

Full bio

Save on Pinterest Save Share on Facebook Share Share on Twitter Tweet
Statistics formulas showing how to calculate the mean, variance, standard deviation, mean absolute deviation, and standard error for a sample data set.

How to Find the Minimum Value

The minimum value is the smallest number in the set. To find the minimum value, start by ordering the numbers from smallest to largest, then simply find the smallest number.

How to Find the Maximum Value

The maximum value is the largest number in the set. Just like finding the minimum value, start by sorting the numbers from smallest to largest, then find the largest number.

How to Find the Count

The count is the size of the data set and is often denoted as n (or N). To find the count, simply count the number of elements in the set.

How to Find the Range

The range is the difference between the minimum value and the maximum value in the data.

\text{range}=\text{max}-\text{min}

Where:
max = highest number
min = lowest number

How to Find the Sum

The sum is the value of each number in the set added together. To find the sum, add up each number to find the total.

\text{sum}=\sum_{i=1}^{n}x_{i}

Where:
xi = each number in the set
n = number of items in the sample

You can also use a tool like our mean, median, mode calculator to find the sum automatically.

How to Find the Mean

The (arithmetic) mean is the average value of a set of numbers and is a measure of the central tendency of the data.

\text{mean}=\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}
\bar{x}=\frac{x_{1}+x_{2}+…+x_{n}}{n}

Where:
x̄ = mean
xi = each number in the set
n = number of items in the sample

How to Find the Median

The median is the middle value in the data set. The median is also a measure of the central tendency of the data. Often, the mean and the median will be the same, as for example in the normal distribution. If the distribution is skewed, however, the mean and median will frequently differ.

For example, mean household incomes are substantially higher than median household incomes, because the salaries of higher-income earners are more disparate than the salaries of lower-income earners.

To find the median, order the data from smallest to largest, then find the middle value in the set. If the number of values in the set is even, then the median is equal to the mean of the middle two values.

Formula for Datasets With Odd Numbers

\text{median}=\left ( \frac{n + 1}{2} \right )^{th}\text{ term}

The median is equal to the (n + 1)/2th term in the dataset, which is equal to the number at the index of the count of the data n plus 1, divided by 2.

Formula for Datasets With Even Numbers

\text{median}=\frac{\left ( \frac{n}{2} \right )^{th}\text{ term} + \left ( \frac{n}{2} + 1 \right )^{th}\text{ term}}{2}

Where:
n = number of items in the set

The median is equal to the n/2th term plus the (n/2) + 1th term, divided by 2.

How to Find the Mode

The mode is the value that occurs most often in the data set. It’s possible for a data set to have multiple modes, if multiple values repeat the same number of times.

If every value in a dataset is unique, then every value is a mode. Informally, some analysts may, in this context, say that the data has no mode, although this is imprecise.

To find the mode, document the frequency that each value occurs in the data. The mode will be the number with the greatest frequency of occurrence.

\text{mode }= L + \left ( \frac{f_{1}-f_{m}}{2f_{1}-f_{m}-f_{2}} \right ) \times h

Where:
L = lower limit of the modal class
fm = frequency of the modal class
f1 = frequency of the class preceding the modal class
f2 = frequency of the class succeeding the modal class
h = size of the class interval

How to Find the Standard Deviation

The standard deviation is a measure of the dispersion or variance between numbers in a data set.

s=\sqrt{\frac{\sum \left ( x_{i}-\bar{x} \right )^{2}}{n-1}}

Where:
x̄ = sample mean
xi = each number in the set
n = number of items in the sample

How to Find the Variance

The variance is the measure of the variability from the mean in a data set. The variance is equal to the standard deviation squared.

s^{2}=\frac{\sum \left ( x_{i}-\bar{x} \right )^{2}}{n-1}

Where:
x̄ = sample mean
xi = each number in the set
n = number of items in the sample

How to Find the Sum of Squares

The sum of squares is a measure of the deviation from the mean for numbers in a data set. It’s often used to calculate variance and standard deviation.

SS = \sum \left ( x_{i}-\bar{x} \right )^{2}

Where:
x̄ = sample mean
xi = each number in the set

How to Find the Quartiles

Quartiles mark the boundaries or divisions of a data set into four equally sized groups.

The first quartile is the median of the lower half of the data, while the third quartile is the median of the upper half.

  • Q1 – first quartile – 25th percentile of the data
  • Q2 – second quartile – 50th percentile of the data
  • Q3 – third quartile – 75th percentile of the data

How to Find the Interquartile Range

The interquartile range is the difference between the first and third quartiles. It is a measure of variability, spread, and dispersion, however it is less sensitive to outliers than other measures that might be chosen.

IQR=Q_{3}-Q_{1}

Where:
Q1 = first quartile
Q3 = third quartile

How to Find the Midrange

The midrange is the arithmetic mean of the smallest and largest numbers in a data set.

\text{midrange}=\frac{\text{min}+\text{max}}{2}

Where:
min = lowest number
max = highest number

How to Find the Mean Absolute Deviation

The mean absolute deviation is the average absolute difference between each value in the set and the mean.

MAD = \frac{\sum|x_{i}-\bar{x}|}{n}

Where:
x̄ = sample mean
xi = each number in the set
n = number of items in the sample

How to Find the Geometric Mean

The geometric mean is the average of a data set that is found using the nth root of the product of each number in the set, where n is the size of the set.

Alternatively, you might regard it as an arithmetic average “on the log scale”, as it is the exponentiated arithmetic average of the logarithms of each number in the set.

\tilde{x} = \left ( \prod_{i=1}^{n}x_{i} \right )^{\frac{1}{n}}
\tilde{x} = \sqrt[n]{x_{1} \cdot x_{2} \cdot\cdot\cdot x_{n}}

Where:
xi = each number in the set
n = number of items in the sample

How to Find the Coefficient of Variation

The coefficient of variation is a measure of relative variability or dispersion of data around the mean in a sample or population. It is formally defined as the ratio of the standard deviation to the mean. In a sample, it may be calculated as follows:

CV=\frac{s}{\bar{x}}\times 100\%

Where:
s = standard deviation
x̄ = sample mean

How to Find the Relative Standard Deviation

The relative standard deviation is a measure of how closely the data is clustered around the mean in a sample or population. It is closely related to the coefficient of variation, although the former can be positive or negative while the latter can only be positive.

RSD=\left |\frac{\sigma}{\mu}\right | \times 100\%

Where:
σ = standard deviation
μ = population mean

How to Find the Standard Error

The standard error is the standard deviation of a sample mean from the actual population mean.

SE=\frac{s}{\sqrt{n}}

Where:
s = standard deviation
n = number of items in the sample

Recommended Statistics Resources

Statistics Calculators Math Calculators