Concepts of Statistics

Statistics is the branch of mathematics that deals with collecting, organising, analysing, interpreting, and presenting data.

In real life, data are everywhere:

• A school analyses students’ marks to understand performance.

• A company studies monthly sales to plan production.

• A doctor records patients’ body temperatures to identify a typical value.

• A government uses census data to plan schools, hospitals, and transport.

Often, a large data set cannot be understood by looking at every value. We therefore use a single representative value called a measure of central tendency.

\[ \begin{array}{c} \text{The mean of statistical data} = \dfrac{\text{The sum of all scores}}{\text{Total no. of scores}} = \dfrac{\sum\limits_{i = 1}^{N} x_{i}}{N} \end{array} \]

(Here \[ x_{i} \] is the \[ i^{th} \] score)

Mean is denoted by \[ \overline{X} \] and it represents the average of the given data.

\[ \overline{X} = \frac{\sum\limits_{i = 1}^{N} x_{i}}{N} \]

\[ \begin{array}{|l|l|} \hline \textbf{SYMBOL} & \textbf{MEANING} \\ \hline \Sigma \text{ (sigma)} & \text{“Add up”: the sum of all the terms that follow.} \\ \hline x_{i} & \text{The } i^{th} \text{ score: } x_{1} \text{ is the first score, } x_{2} \text{ the second, and so on.} \\ \hline i = 1 \text{ (below } \Sigma) & \text{Start adding from the first score.} \\ \hline N \text{ (above } \Sigma) & \text{Stop at the last score; } N \text{ is the total number of scores.} \\ \hline \overline{X} \text{ (X-bar)} & \text{The mean (average) of the data.} \\ \hline \end{array} \]

Expanded:

\[ \sum\limits_{i = 1}^{N} x_{i} = x_{1} + x_{2} + x_{3} + \ldots + x_{N} \]

Imagine three friends want to describe the “typical” pocket money in a group.

• One friend calculates the mean by adding everyone’s pocket money and dividing by the number of students.

• Another finds the median by arranging the amounts and choosing the middle value.

• A third finds the mode by identifying the pocket-money amount that most students receive.

All three are correct, but they describe the data in different ways.

• Statistics deals with collecting, organising, analysing, and interpreting data.

• Central tendency gives a single representative value for a data set.

• Mean is the arithmetic average.

• Median is the middle value of ordered data.

• Mode is the most frequently occurring value.