Statistics Formulas
The branch of Statistics deals with the study of the collection, analysis, interpretation, presentation, and organization of data. Statistics formulas are used to analyze the data and help to interpret various results and presume possibilities. Let us learn about the basic statistics formulas with a few examples at the end.
What are the Statistics Formulas?
The basic statistics formulas help us in interpreting the facts, and information that is in the form of numeric data only. With the help of statistics formulas, we are able to find various measures of central tendencies and the spread/deviation of data values from the center. The important statistics formulas are listed below:
\(\begin{array}{cc} \hline \text { Mean } (\bar{x}) & \bar{x}=\frac{\sum x}{n} \\ \hline \text { Median (M)} & \begin{array}{c} \text { If } \mathrm{n} \text { is odd, then } \\ \mathrm{M}=\left(\frac{n+1}{2}\right)^{t h} \text { term } \\ \text { If } \mathrm{n} \text { is even, then } \\ \mathrm{M}=\frac{\left(\frac{n}{2}\right)^{t h} \text { term }+\left(\frac{n}{2}+1\right)^{t h} \text { term }}{2} \end{array} \\ \hline \text { Mode } & \text { The value which occurs most frequently } \\ \hline \text { Variance }(\sigma^{2}) & \sigma^{2}=\frac{\sum(x\bar{x})^{2}}{n} \\ \hline \text { Standarad Deviation } (S)& S=\sigma=\sqrt{\frac{\sum(x\bar{x})^{2}}{n}} \\ \hline \end{array}\)
where,
 x = Observations given
 \(\bar{x}\) = Mean
 n = Total number of observations
Let us now have a look at a few solved examples using the statistics formulas.
Solved Examples on Statistics Formulas

Example 1: Age of students = {14,15,16,15,17,15,18}. Find the mode.
Solution:
Since there is only one value repeating itself, it is a unimodal list.
\(\begin{align} \text{Mode} = \left \{ {15} \right \}\end{align}\)
Answer: Mode = 15

Example 2: In a class of 50, 4 students were selected at random and their total marks in the final assessments are recorded, which are: 812, 836, 982, 769. Find the standard deviation of their marks using statistics formula. (Take √23.1= 4.8)
Solution:
N = 4
Sample Mean (X̄) = \( \dfrac{812+836+982+769}{4} \) = 849.75
Variance = \( \dfrac{\sum^{N}_{i=1} (X_i  \bar{X})^2}{N} \)
=\( \dfrac{\sum^{4}_{i=1} (X_i  849.75)^2}{4} \)
= \( \dfrac{(812  849.75)^2 + (836  849.75)^2 + (982  849.75)^2 + (769  849.75)^2}{4} \) = 6406.1875
Standard Deviation = \(\sqrt{6406.1875}\) = 80.039
Answer: Standard Deviation for this data is 80.039.
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