Economics · Ch 11 — Measures of Central Tendency

Recap

Central tendency is a single value that represents the centre of a data set. The three main measures are the mean, median, and mode.
Arithmetic mean ( $\bar{x}$ ) for ungrouped data: $\bar{x} = \frac{\sum x_i}{n}$ . For grouped data: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ (direct method) or $\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i}$ (step-deviation method, where $d_i = \frac{x_i - A}{h}$ ). It is affected by extreme values.
Median is the middle value when data is arranged in order. For ungrouped data with $n$ values, median = $\left(\frac{n+1}{2}\right)^{\text{th}}$ term. For grouped data: $\text{Median} = L + \frac{\frac{N}{2} - cf}{f} \times h$ , where $L$ is the lower limit of the median class, $cf$ is cumulative frequency before it, $f$ is its frequency, $h$ is class width, and $N = \sum f_i$ . It is not affected by extreme values.
Mode is the most frequent value. For grouped data: $\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$ , where $f_1$ is frequency of modal class, $f_0$ and $f_2$ are frequencies of preceding and succeeding classes. For symmetric distributions, mean = median = mode; for skewed data, an empirical relation is $\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}$ . …