Measures of Central Tendency and Dispersion

1) Why these measures matter

Statistical methods summarize data to understand:

• the “typical” value (central tendency),
• the spread/variability (dispersion), which are essential for economics data interpretation.

2) Measures of Central Tendency

2.1 Arithmetic Mean (AM)

For ungrouped data $x_1, x_2, ..., x_n$:

For frequency distribution:

Merits: uses all values; algebraic properties.
Demerits: affected by extreme values (outliers).

2.2 Median

Median is the middle value when data are arranged in order.

• If $n$ is odd: median is $\left(\frac{n+1}{2}\right)^{th}$ item.
• If $n$ is even: median is average of $\left(\frac{n}{2}\right)^{th}$ and $\left(\frac{n}{2}+1\right)^{th}$ items.

Merits: robust to outliers; useful for skewed distributions.
Demerits: ignores magnitude of most observations.

Grouped data median (concept)
Median class contains $\frac{N}{2}$th item.

Where:

• $L$ = lower boundary of median class
• $N$ = total frequency
• $c.f.$ = cumulative frequency before median class
• $f$ = frequency of median class
• $h$ = class width

2.3 Mode

Mode is the value that occurs most frequently.

Grouped data mode (concept)
Modal class has highest frequency.

Where:

• $f_1$ = frequency of modal class
• $f_0$ = frequency of class preceding modal class
• $f_2$ = frequency of class succeeding modal class