Business Mathematics and Basic Statistics · Ch 4 — Measures of Central Tendency

Arithmetic Mean — Discrete Data

The arithmetic mean (usually called simply the mean) is the most widely used measure of central tendency — a single representative value for a whole set of data. Whether we are averaging test marks, daily sales, or the number of children in a set of families, the mean answers one simple question: if every observation were replaced by the same number, what would that number have to be so the total stays unchanged?

Note

Arithmetic Mean — Individual Observations

For $n$ individual observations $x_1, x_2, \ldots, x_n$ , the arithmetic mean is

$\bar{x} = \dfrac{x_1+x_2+\cdots+x_n}{n} = \dfrac{\sum x}{n}$

Often, several observations in a data set repeat the same value. Instead of writing a value as many times as it occurs, it is more convenient to record each distinct value once along with the number of times it occurs — its frequency. Data organised this way, as pairs of distinct values $x_i$ and frequencies $f_i$ , is called a discrete frequency distribution (this is different from a grouped or continuous frequency distribution, covered in the next section, where values are bundled into class intervals rather than listed individually).

Note

Arithmetic Mean — Discrete Frequency Distribution

For a discrete frequency distribution with distinct values $x_1, x_2, \ldots, x_n$ occurring with frequencies $f_1, f_2, \ldots, f_n$ respectively,

$\bar{x} = \dfrac{f_1x_1+f_2x_2+\cdots+f_nx_n}{f_1+f_2+\cdots+f_n} = \dfrac{\sum fx}{\sum f}$

where $\sum f = N$ is the total number of observations.

A property worth remembering (it is what makes the assumed-mean shortcut of the next section work): the sum of the deviations of all observations from their own arithmetic mean is always exactly zero, i.e. $\sum (x_i-\bar{x}) = 0$ , and for a frequency distribution, $\sum f(x_i-\bar{x}) = 0$ .

Computing the arithmetic mean of a discrete frequency distribution — both for a plain list of values and for a table of values against frequencies — is one of the most frequently examined skills in the WBCHSE Class 11 Business Mathematics and Basic Statistics semester paper, and the same underlying idea of an "average" is taught under measures of central tendency in commerce and statistics courses across every Indian board.

Definition 1Arithmetic Mean

The sum of all observations divided by the number of observations; $\bar{x} = \sum x / n$ for individual data, $\bar{x} = \sum fx / \sum f$ for a frequency distribution.

Definition 2Discrete Frequency Distribution

A table listing each distinct value $x_i$ a variable takes, together with the number of times ( $f_i$ , its frequency) that value occurs — as opposed to a continuous/grouped distribution, where values are bundled into class intervals.

← Previous chapterCh 3 — Sales Tax, VAT and GST Next · 2 →Arithmetic Mean — Grouped Data