Descriptive Statistics Calculator

A descriptive statistics calculator is used to summarize and describe the main features of a dataset. It provides a range of statistical metrics that offer insights into the data distribution's central tendency, dispersion, and shape.

What are Descriptive Statistics?

Descriptive statistics provide concise summaries of datasets, which can be from a whole population or a sample. These summaries are broken into central tendency measures such as mean, median, and mode and variability, including standard deviation, variance, range, kurtosis, and skewness.

Formulas to Calculate Descriptive Statistics

Descriptive statistics gives a summary of data sets by describing features such as central tendency, dispersion, and shape. Here are the common descriptive statistics and how to calculate them:

1. Measures of Central Tendency

Mean: The average of all data points.

Mean = X₁ + X₂ + X₃ +… + Xn / n

Median: The middle value is when data points are ordered from least to greatest. If there is an even number of observations, the median is the average of the two middle
Mode: Most frequently occurring value(s) in the data set.

2. Measures of Dispersion

Range: The difference between the highest and lowest values.

Range = Maximum − Minimum

Variance: The average of the squared differences from the mean.

Variance = [Σ (x − 𝜎)²/ n-1]

Standard Deviation: The square root of the variance.

Standard Deviation = Square root of [Σ (x − 𝜎)²/ n-1]

Interquartile Range (IQR): The difference between the first quartile (25th percentile) and the third quartile (75th percentile).

IQR =𝑄₃−𝑄₁

3. Measures of Shape

Skewness: A measure of the asymmetry of the data distribution. Positive skewness indicates a distribution with a long tail to the right, while negative skewness indicates a long tail to the left.
Kurtosis: A measure of the “tailedness” of the data distribution. High kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations.

How to Calculate Descriptive Statistics?

To calculate the Descriptive statistics, you need to follow these simple steps:

Gather the data points you need to analyze
Arrange the data in ascending order if necessary.
Compute the mean, median, and mode.
Compute the range, variance, standard deviation, and IQR.
Compute skewness and kurtosis if necessary.

Examples of Descriptive Statistics

This section shows how descriptive statistics can summarize and analyze real-life data sets. It also offers information on the central tendency, dispersion, and distribution of the data.

Example:

A school wants to understand the performance of its students in a recent exam. The scores of the students are as follows: 56,78,89,92,67,73,85,90,88,76. Calculate the Descriptive Statistics.

Solution

Measure of Central Tendency

Mean =56+78+89+92+67+73+85+90+88+76 / 10 =794 / 10=79.4
Median: First, arrange the data in ascending order: 56,67,73,76,78,85,88,89,90,92

Since there are 10 data points, the median is the average of the 5th and 6th values:

Median=78+85/ 2 = 81.5

Mode: The value that appears most frequently: In this data set, all values are unique, so there is no mode.

Measure of Dispersion

Range = 92−56 =36

Variance = First, calculate the mean (μ=79.4), then find the squared differences from the mean and average them:

X_i	X_i - X	(X_i- X)²
56	-23.400	547.56
67	-12.400	153.76
73	-6.400	40.96
76	-3.400	11.56
78	-1.400	1.96
85	5.599	31.36
88	8.599	73.96
89	9.599	92.16
90	10.599	112.36
92	12.599	158.76
--	--	∑ (Xi - X)² = 1224.39

Putting calculated values in the formula:

Variance = [1/10 (1224.4)]

Variance = (0.1) (1224.4) = 122.44

Standard Deviation: σ = 11.065
Quartile: Q1 --> 73>>Q2 --> 81.5>>Q3 --> 89

Interquartile Range (IQR) = 16
Sum of Squares = 1224.400
Mean Absolute Deviation = 9.400
Root Mean Square = 80.167
Std Error of Mean = 3.499

Measure of Shape

Skewness = -0.748
Kurtosis = 2.522
Kurtosis Excess = -0.478
Coefficient of Variation = 0.139
Relative Standard Deviation = 13.936

Frequency Table:

Value	Frequency	Frequency %
56	1	10.00%
67	1	10.00%
73	1	10.00%
76	1	10.00%
78	1	10.00%
85	1	10.00%
88	1	10.00%
89	1	10.00%
90	1	10.00%
92	1	10.00%

Frequently Asked Questions

What are the main types of Descriptive Statistics?

There are three main types of descriptive statistics:

Frequency Distribution.
Central Tendency.
Variability of Data set.

What is the purpose of calculating descriptive statistics?

There are two main purposes for calculating descriptive statistics:

Providing basic information about variables in a dataset.
Highlighting potential relationships between variables.

What is the difference between descriptive statistics and inferential statistics?

The main difference between both of them is that:

Descriptive statistics focus on describing the visible characteristics of a dataset (a population or sample).
Inferential statistics focus on making predictions or generalizations about a larger dataset, based on a sample of those data.