Statistics Formulas

Last Updated : 22 Feb, 2024

Statistics is a branch of mathematics that deals with numerical data analysis. It presents the data in an organized manner. Statistics is a study of a collection of data, analysis it, interpretation, and presentation of data in a well-organized form. It allows us to interpret various results from the given data and forecast many possibilities. The theory of statistics defines a function of a sample where the function is completely independent of the sample’s distribution. Statistics helps to find various measures of central tendencies and the deviation of different observations from the center. Let’s discuss the statistics formulas that find the measures of central tendencies and deviations.

Statistics Formulas

Statistics formulae include mean, median, mode as well as standard deviation. Mean tells the average of the data, median tells the middle of the center value of the data, median tells the most frequent value in the data. Let’s learn about these terms and their formulae in more detail,

Mean

Mean is one of the measures of central tendency. It finds the average value for the given data/observations. Arithmetic mean is defined as the sum of all the numbers in the data divided by the total count of numbers. The formula for finding the mean is given by,

Arithmetic mean ( $\bar{x}$ ) = ∑x/n

Where ∑x is summation of all observations.

n represents total count of all numbers/observations.

Median

Median is also one of the measures of central tendency. It gives the middle value in the given ordered data. The formula for finding the median is given by,

Median = [(n + 1)/2]^th term

Where n is the total count of numbers/observations.

The above formula is applicable only when n is odd.

If n is even then median is calculated by the formula

Median = [(n/2)^th term + [(n/2) + 1]^th term]/2

Note: The above formulas can be applied only when the data is ordered. So before calculating median the data should be ordered either in ascending or descending order.

Mode

Mode specifies the most repeating element in the given data. It specifies the value that occurs mostly.

Variance

Variance measures the variability of the given data from the mean. It is the expectation of the squared deviation of a random variable from its sample mean. Variance is equal to the square of standard deviation. The formula for calculating variance is given by,

Variance (σ²) = $\frac{∑(x - \bar{x})^2}{n}$

Where x is the observations given

$\bar{x}$ is the mean of the given data

n represents total count of observations.

Standard Deviation

Standard deviation measures the amount of variation/dispersion of a set of values. Dispersion tells how much data is spread out. A lower standard deviation indicates that data is close to the center. The higher value of standard deviation represents that data spread is more.

Standard Deviation (σ) = $\sqrt{\frac{∑(x-\bar{x})^2}{n}}\sqrt{\frac{∑(x-\bar{x})^2}{n}}$

Standard Deviation = √{Variance}

Sample Problems

Question 1: Find the mean for the given data 10, 20, 60, 40, 25, 35

Solution:

Given data,

10, 20, 60, 40, 25, 35

n = 6

Arithmetic mean ( $\bar{x}$ ) = ∑x/n

= (10 + 20 + 60 + 40 + 25 + 35)/6

= 190/6

= 31.66

Mean for the given data is 31.66

Question 2: Find the median for the given data 10, 20, 60, 40, 25, 35

Solution:

Given data is not ordered. So in order to calculate median value the data should be ordered.

Here the given data is ordered in ascending order.

10, 20, 25, 35, 40, 60

n = 6

n is even, median formula is,

Median = [(n/2)^th term + [(n/2) + 1]^th term ]/2

= [(6/2)^th term + [(6/2) + 1]^th term]/2

= (3^rd term + 4^th term)/2

= (25 + 35)/2

= 30

Median for the given data is 30.

Question 3: Find the median for the given data 10, 20, 60, 40, 25, 35, 50

Solution:

Given data is not ordered. So in order to calculate median value the data should be ordered.

Here the given data is ordered in ascending order.

10, 20, 25, 35, 40, 50, 60

n = 7

n is odd, median formula is,

Median = [(n + 1)/2]^th term

= [(7 + 1)/2]^th term

= 4^th term

= 35

Median for the given data is 35.

Question 4: Find the mode for the data 1, 2, 2, 2, 3, 3, 4

Solution:

Here the most repeated value is 2 which occurred three times.

So the mode for the given data is 2.

Question 5: Find the variance for the data 1, 2, 5, 4, 8, 4

Solution:

Given data,

1, 2, 5, 4, 8, 4

n = 6

Arithmetic mean ( $\bar{x}$ ) = ∑x/n

= (1 + 2 + 5 + 4 + 8 + 4)/6

= 24/6

= 4

$\bar{x}$ = 4

Variance (σ²) = $\frac{∑(x-\bar{x})^2}{n}$

= [(1 – 4)²+ (2 – 4)²+ (5 – 4)²+ (4 – 4)²+ (8 – 4)²+ (4 – 4)²]/6

= (9 + 4 + 1 + 0 + 16 + 0)/6

= 30/6

= 5

Variance for the given data is 5.

Question 6: Find the variance for the data 1, 2, 5, 4, 8

Solution:

Given data,

1, 2, 5, 4, 8

n = 5

Arithmetic mean ( $\bar{x}$ ) = ∑x/n

= (1 + 2 + 5 + 4 + 8)/5

= 20/5

= 4

$\bar{x}$ = 4

Standard Deviation (σ) = $\sqrt{\frac{∑(x-\bar{x})^2}{n}} = \sqrt{\frac{(1-4)^2+(2-4)^2+(5-4)^2+(4-4)^2+(8-4)^2}{5}} = \sqrt{\frac{9+4+1+0+16}{5}} = \sqrt{\frac{30}{5}}$

= √6

Standard deviation for the given data is 2.45