# Descriptive Statistics Calculator

## Calculator Use

Find descriptive statistics of a data set. Descriptive statistics describe the main features of a data set in quantitative terms. This calculator will generate certain descriptive statistics for a sample data set with 4 or more values and up to 5000 values. Enter values separated by commas such as 1, 2, 4, 7, 7, 10, 2, 4, 5.

You can also copy and paste lines of data points from documents such as Excel spreadsheets or text documents in the following formats with or without commas:

Unit

Options

54

65

47

59

40

53

54,

65,

47,

59,

40,

53,

or

42, 54, 65, 47, 59, 40, 53

65 47

59 40

53

or

42 54 65 47 59 40 53

54 65,,, 47,,59,

40 53

Below is a listing of the statistical values calculated and generally how they are determined with this calculator. Values in a set of data are represented by x1, x2, x3, ... xn.

## Descriptive Statistics Formulas and Calculations

- Minimum
- The smallest value in a sample data set.
- Maximum
- The largest value in a sample data set.
- Range
- The range from the minimum to the maximum;

*range = max - min* - Sum
- The total of all data values.

(x1 + x2 + x3 + ... + xn)\( \text{Sum} = \sum_{i=1}^{n}x_i \) - Count (n)
- The total number of data values in a data set.
- Mean
- The sum of all of the data divided by the count; the average;

*mean = sum / n*.\( \text{Mean} = \dfrac{\sum_{i=1}^{n}x_i}{n} \) - Median
- The numeric value separating the higher 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.
- Mode
- The value or values that occur most frequently in the data set.
- Standard Deviation (s)
- The square root of the variance;

^{2}√variance or variance = s^{2}\( \text{s} = \sqrt{\dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1}} \) - Variance (s
^{2}) - The sum of the squared differences between data values and the mean, divided by the count - 1;

[ (x1 - mean)^{2}+ (x2 - mean)^{2}+ (x3 - mean)^{2}+ ... + (xn - mean)^{2}] / [n - 1]\( \text{s}^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1} \) - Midrange
- The average of the minimum and maximum of the data set. (min + max)/2
- Quartiles
- The median is the second quartile Q
_{2}. It divides the ordered data set into higher and lower halves. The first quartile, Q_{1}, is the median of the lower half not including Q_{2}. The third quartile, Q_{3}, is the median of the higher half not including Q_{2}. This is 1 of several methods for calculating the quartiles.^{[1]} - Interquartile Range
- The range from Q
_{1}to Q_{3}is the interquartile range (IQR).

[Q_{3}- Q_{1}] - Outliers
- Potential outliers within the sample data set. Outliers are values that lie above the Upper Fence or below the Lower Fence of the sample set.

Upper Fence = Q_{3}+ 1.5 × Interquartile Range

Lower Fence = Q_{1}− 1.5 × Interquartile Range - Sum of Squares
- The sum of the squared differences between data values and the mean;

[ (x1 - mean)^{2}+ (x2 - mean)^{2}+ (x3 - mean)^{2}+ ... + (xn - mean)^{2}]\( = \sum_{i=1}^{n}(x_i - \overline{x})^{2} \) - Mean Deviation (Mean Absolute Deviation)
^{[2]} - The sum of the absolute of the differences between data values and the mean, divided by the count;

[ |x1 - mean| + |x2 - mean| + |x3 - mean| + ... + |xn - mean| ] / n\( = \dfrac{\sum_{i=1}^{n}|x_i - \overline{x}|}{n} \) - Root Mean Square (RMS)
- The square root of, the sum of the squared data values divided by n

^{2}√[ (x1^{2}+ x2^{2}+ x3^{2}+ ... + xn^{2}) / n ]\( = \sqrt{\dfrac{\sum_{i=1}^{n}x_i^{2}}{n}} \) - Standard Error of the Mean
- The standard deviation divided by the square root of the count;

s /^{2}√n - Skewness
^{} - The sum of the cubed differences between data values and the mean, divided by the count minus 1 times the cubed standard deviation;

[n/((n - 1)(n - 2))][ (x1 - mean)^{3}+ (x2 - mean)^{3}+ (x3 - mean)^{3}+ ... + (xn - mean)^{3}] / [ s^{3}]\( = \dfrac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\dfrac{x_i - \overline{x}}{s}\right)^{3} \) - Kurtosis
^{} - The sum of the fourth power of differences between data values and the mean, divided by the count minus 1 times the fourth power of the standard standard deviation;

[n(n + 1)/((n - 1)(n - 2)(n - 3))][ (x1 - mean)^{4}+ (x2 - mean)^{4}+ (x3 - mean)^{4}+ ... + (xn - mean)^{4}] / [ s^{4}]\( = \dfrac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^{n} \left(\dfrac{x_i - \overline{x}}{s}\right)^{4} \) - Kurtosis Excess (Excel's Kurtosis)
- The sum of the fourth power of differences between data values and the mean, divided by the count minus 1 times the fourth power of the standard standard deviation;

[n(n + 1)/((n - 1)(n - 2)(n - 3))][ (x1 - mean)^{4}+ (x2 - mean)^{4}+ (x3 - mean)^{4}+ ... + (xn - mean)^{4}] / [ s^{4}] - [ (3(n - 1)^{2}) / ((n-2)(n-3)) ]\( = \dfrac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^{n} \left(\dfrac{x_i - \overline{x}}{s}\right)^{4} \)\( - \dfrac{3(n-1)^{2}}{(n-2)(n-3)} \) - Coefficient of Variation
- The standard deviation divided by the mean;

s / mean - Relative Standard Deviation (expressed as a percentage)
- 100 times the standard deviation divided by the mean;

(100 * s / mean)% - Frequency
- The number of occurrences for each data value in the data set.

### References

[1] Wikipedia contributors. "Quartile." Wikipedia, The Free Encyclopedia. Last visited 11 July, 2016.

[2] Weisstein, Eric W. "Mean Deviation." From MathWorld--A Wolfram Web Resource. Mean Deviation.

[3] Information Technology Lab, National Institute of Standards and Technology. Section 1.3.5.11 Measures of Skewness and Kurtosis. From the Engineering Statistics Handbook.

**Cite this content, page or calculator as:**

Furey, Edward "Descriptive Statistics Calculator"; CalculatorSoup,
*https://www.calculatorsoup.com* - Online Calculators