What is Descriptive Statistics?

Descriptive Statistics is a method of organizing, summarizing, and presenting data in a convenient and informative way.
The actual method used depends on what information we would like to extract.

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Areas of Interest for Descriptive Statistics

MEASURES OF CENTRAL TENDENCY
MEAN (Arithmetic Average):
Mean is the arithmetic average computed by summing all the values in the dataset and dividing the sum by the number of data values. For a finite set of dataset with measurement values X1, X2, …., Xn (a set of n numbers), it is defined by the formula:

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Mean Formula

The sample mean is represented by x-bar.
The population mean is represented by Greek letter µ.

For a given data set: 12, 14, 11, 12, 12, 12, 15, 17, 22, 15, 12
Sum of data points = (12+14+11+12+12+12+15+17+22+15+12) = 154
Number of data points = (take a total count of observations) = 11
Mean = (Divide the sum of data points by the total number of data points) = 154/11 = 14


MEDIAN:
The median is the middle value after the data are sorted from smallest to largest. With an odd number n of observations, it is the value in position (n + 1)/2. With an even number, it is the average of the values in positions n/2 and (n/2) + 1.

For a given data set: 12, 14, 11, 12, 12, 12, 15, 17, 22, 15, 12
Ascending Order: 11, 12, 12, 12, 12, 12, 14, 15, 15, 17, 22
Thus, the middle number in the data set Median = 12


MODE:
Mode is the data point having the highest frequency (maximum occurrences).

For a given data set: 12, 14, 11, 12, 12, 12, 15, 17, 22, 15, 12
Maximum occurring data point, Mode = 12


QUARTILES:
Quartiles are the 25th, 50th and 75th percentiles of ordered data. They mark four portions of the distribution; with a finite sample or tied values, these need not contain exactly equal numbers of observations. Sample-quartile conventions can differ.

• First quartile (designated Q1) = lower quartile = cuts off lowest 25% of data = 25th percentile
• Second quartile (designated Q2) = median = cuts data set in half = 50th percentile
• Third quartile (designated Q3) = upper quartile = cuts off highest 25% of data, or lowest 75% = 75th percentile
• The difference between the upper and lower quartiles is called the interquartile range.

MEASURES OF CENTRAL DISPERSION/VARIATION
STANDARD DEVIATION:
Standard deviation measures spread using squared deviations from the mean, followed by a square root; it is not the simple average of absolute distances. Population standard deviation is denoted by σ, and sample standard deviation by s.

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Standard Deviation Formula

In the sample-standard-deviation formula, Sx denotes the sample standard deviation.
xi is an observation, x-bar is the sample mean, and n is the sample size.
Σ means sum the squared deviations (xi − x-bar)² for all observations. Divide this sum by n − 1, then take the square root.


VARIANCE:
Variance is defined as the square of standard deviation. Variance of the population is represented as σ times σ. Variance for the sample is represented as "s times s".

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Variance Formula

In the sample-variance formula, Sx² denotes the sample variance.
xi is an observation, x-bar is the sample mean, and n is the sample size.
Σ means sum the squared deviations (xi − x-bar)² for all observations. Divide this sum by n − 1.


RANGE:
Range is defined as the difference between largest value in a data set and the smallest value in a data set.

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Range Formula

ValueMax stands for the highest (maximum) value in the data set and ValueMin stands for the lowest (minimum) value in the data set.

In a given data-set like 12, 13, 11, 12, 12
Range: 13 – 11 = 2
Mean: (12+13+11+12+12) / 5 = 12
Variance: Sum of [(X – mean) times (X – mean)] / (n – 1) = [0+1+1+0+0] / (5 – 1) = 2 / 4 = 0.50
Standard Deviation: Square Root of 0.50 = 0.7071