The Gap Between Mean and Median — medium Descriptive Statistics problem | Incognition

The Gap Between Mean and Median

Descriptive StatisticsMedium+20 XP+20

The Gap Between Mean and Median

A call centre records how many calls each of its $40$ agents handled during one shift. Instead of listing $40$ numbers, the supervisor tallies them:

Calls handled ( $x$ )	Number of agents ( $f$ )
$4$	$6$
$5$	$14$
$6$	$10$
$7$	$6$
$8$	$4$

Compute the mean number of calls and the median number of calls for these $40$ agents, then report

$\text{mean} - \text{median}$

Round your answer to two decimal places.

Hints

calls

Weighted Mean Lab

Explore how values and their corresponding weights combine to pull the average.

Frequency Weights

Observation Value (x)	Frequency / Weight (w)	Weighted Contribution (x × w)
Totals:	Sum of Freqs = 0	Sum of Products = 0

Method 1: Expanded Arithmetic Mean

Write out every value and take their ordinary average:

Values: [2, 2, 2, 5, 5, 8]
Mean = (2+2+2+5+5+8) / 6 = 24/6 = 4.0

Method 2: Weighted Mean using Frequencies

Multiply each unique value by its frequency and divide by total count:

Mean = (2×3 + 5×2 + 8×1) / (3+2+1)
= (6 + 10 + 8) / 6 = 24/6 = 4.0

Frequency is a Weight: By using frequency as a weight, we do not have to write out thousands of raw, repeated observations. The weighted mean gives us the exact same result as averaging the full, expanded dataset, which makes it extremely useful in grouping big data!

The Gap Between Mean and Median