Data in Mathematics
2026-02-28 09:16 Diff
https://brightchamps.com/en-us/math/data

Data analysis techniques are used to organize and examine data to get relevant information about a given problem. These techniques help identify patterns and refine raw information given by the data. Here are some of the data analysis techniques that are often used.

1. Mean -  The mean is used to find the average of a set of numbers. To calculate the mean, add all the values together and divide the total by the number of values. The formula to find the mean is: 


\(σ = \frac{Sum \ of \ given \ data}{Total \ number \ of\ data}\).
 

Let’s understand this with an example and try to find the mean for the exam scores, 85, 90, 68, 95, and 55.


Use the formula to find the average score. Substituting the scores in the formula, we get:


\(Mean=\frac{(85 + 90 + 68 + 95 + 55)}{5}\)


\(Mean = \frac{393}{5}\)


\(Mean= 78.6\)


2. Median - Median is the middle value in a dataset. We have to arrange the numbers in ascending or descending order and take the middle number as the median. If the given numbers are even, take the average of the two middle numbers.

Example:
 

(i) 1, 2, 3, 4, and 5. Here, the median value is 3.
 

(ii) 1, 2, 3, and 4. Here, we will find the median value using the following formula.
 

\(Median = \frac{sum \ of \ the \ middle \ two \ numbers}{2}\)
 

\(Median = \frac{(2 + 3)}{2} = 2.5\)

3. Mode - The mode is the value that appears more frequently in a dataset. Unimodal, bimodal, and multimodal are some of the types of modes. If the dataset contains only one mode, it is called unimodal. If there are two modes, it’s called bimodal, and if there are more than two modes, it is called multimodal. A dataset with no repeated values has no mode.

Example: Find the mode in the dataset 2, 5, 7, 6, 5, 7, 8.
 

The modes in the given dataset are 5 and 7 because they appear twice in the dataset. The dataset has two modes and hence it is bimodal.

4. Range - The difference between the lowest value and the highest value in a dataset is the range. The range is calculated by using the formula:

\(Range = Highest \ value - Lowest \ value\).

Example: Find the range using the dataset 89, 76, 65, 78, 54, 23.

Here, the highest value = 89

\(Lowest \ value = 23\)

\(Range = Highest \ value - Lowest \ value\)

\(Range= 89 - 23\)

\(Range = 66\)

5. Standard deviation formula - The amount of variation in the given dataset is known as the standard deviation. We can find the standard deviation using the formula: 

\(\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}} \)

Here, xi is every value in the dataset.

μ is the mean value in the dataset.

N is the given number of values.

Example: Find the standard deviation for 1, 2, 3, with the mean 2.

Standard Deviation 

\(\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}} \)

Let's find the deviations from mean

  • \(1 − 2 = −1\)
     
  • \(2 − 2 = 0\)
     
  • \(3 − 2 = 1\)

Squaring the deviations, we get:

  • \((−1)^2 = 1\)
     
  • \(0^2 = 0\)
     
  • \(1^2 = 1\)

Adding them, we get:
 

\(1 + 0 + 1 = 2\)

Dividing by N,
 

\(\frac{2}{3} ≈ 0.6667\).

Taking the square root,
 

\( = 0.6667 ≈ 0.8165 ​\)

6. Variance - Variance measures the spread of data. In math, it is used to find the square of the difference in the mean. The formula used to find the variance is as follows: 


\( \sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n} \)
 

Here,

xi is every value in the dataset.

\(μ\) is the mean value in the dataset.

n is the number of values.

Example: Find the variance of 1, 2, 3 with mean as 2

\(\text{Variance}= \frac{((1 - 2)^2 + (2 - 2)^2 + (3 - 2)^2)}{3}\).

\(\text{Variance}= \frac{(1 + 0 + 1)}{3}.\)

\(\text{Variance}= \frac{2}{3}\)

\(\text{Variance}≈ 0.667\)