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# Descriptive Statistics Calculator

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:

Acceptable Delimited
Data Formats
Type
Unit
Options
Actual Input Processed
Column (New Lines)
42
54
65
47
59
40
53
42, 54, 65, 47, 59, 40, 53
Comma Separated (CSV)
42,
54,
65,
47,
59,
40,
53,

or

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

or

42 54 65 47 59 40 53
42, 54, 65, 47, 59, 40, 53
Mixed Delimiters
42
54   65,,, 47,,59,
40 53
42, 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 = s2
$$\text{s} = \sqrt{\dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1}}$$
Variance (s2)
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 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 1 of several methods for calculating the quartiles.[1]
Interquartile Range
The range from Q1 to Q3 is the interquartile range (IQR).
[Q3 - Q1]
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 = Q3 + 1.5 × Interquartile Range
Lower Fence = Q1 − 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√[ (x12 + x22 + x32 + ... + xn2) / 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 ] / [ s3 ]
$$= \dfrac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\dfrac{x_i - \overline{x}}{s}\right)^{3}$$
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 ] / [ s4 ]
$$= \dfrac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^{n} \left(\dfrac{x_i - \overline{x}}{s}\right)^{4}$$
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 ] / [ s4 ] - [ (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