Descriptive Statistics

Lessons

Senior 4 · Basic Mathematics

Descriptive Statistics

5 lessons

Lesson 3 of 5

Mode, mean and median of ungrouped and grouped data

25 min

How big is a "typical" district?

Add up the 30 district populations of 2022 and divide by 30: the answer is about 441 500. Yet 20 of the 30 districts have fewer people than that. The middle district, by contrast, has about 400 500. Which number describes a typical district? The answer depends on the shape of the data, and on one district in particular: Gasabo, with 879 505 residents.

Definition

Mean (arithmetic mean): the sum of the values divided by their number:

\bar{x} = \frac{\sum x}{n} \qquad \text{or, from a frequency table,} \qquad \bar{x} = \frac{\sum f x}{\sum f} .

Median: the middle value when the data are in order; for $n$ values it is at position $\frac{n + 1}{2}$ (the mean of the two middle values when $n$ is even). Mode: the value (or class) with the highest frequency; a data set may have one mode, several, or none.

Grouped data: estimates

Once data are grouped, the exact values are lost, so the averages can only be estimated.

Mean: treat every value in a class as its midpoint $x$ : $\bar{x} \approx \frac{\sum f x}{\sum f}$ .
Median: find the class containing the $\frac{n}{2}$ th value (the median class), then interpolate, assuming the values are spread evenly through it: $\text{Median} \approx L + \frac{\frac{n}{2} - F}{f_m} \times c$ where $L$ is the lower boundary of the median class, $F$ the cumulative frequency before it, $f_{m}$ its frequency and $c$ its width.
Mode: the modal class has the highest frequency (or frequency density); within it, $\text{Mode} \approx L + \frac{d_1}{d_1 + d_2} \times c$ where $d_{1}$ and $d_{2}$ are the differences between the modal frequency and the frequencies of the classes before and after it.

The dot plot shows why the two averages differ. The median sits in the thick of the dots; the mean is pulled to the right by the long tail. In a right-skewed distribution, mode $< \lt$ median $< \lt$ mean.

Tip

Mean for totals, median for typical, mode for most common. If you will multiply back up to a total (food for all districts), use the mean. If you want a "typical" member of skewed data (incomes, house prices, district sizes), use the median. For qualitative data, or the size a shop should stock most of, use the mode.

✦

Worked example 1: averages from a frequency table

Find the mean, median and mode of the household sizes from lesson 1 (illustrative data).

$x$	1	2	3	4	5	6	7	Total
$f$	2	3	4	5	3	2	1	20
$f x$	2	6	12	20	15	12	7	74
Cumulative $f$	2	5	9	14	17	19	20

\bar{x} = \frac{\sum f x}{\sum f} = \frac{74}{20} = 3.7\ \text{people}

Median: $n = 20$ , so it is the mean of the 10th and 11th values. The cumulative frequency passes 9 at size 3 and reaches 14 at size 4, so both are 4: median $= 4$ . Mode: 4 (frequency 5).

Worked example 2: the census data, and what one district does

The 30 district populations add up to the national total, 13 246 394.

\bar{x} = \frac{13\,246\,394}{30} \approx 441\,546

In order, the 15th and 16th values are Gisagara (397 051) and Ngoma (404 048):

\text{Median} = \frac{397\,051 + 404\,048}{2} = 400\,549.5

All 30 values are different, so there is no mode. Without Gasabo the 29 districts have mean $\frac{13\,246\,394 - 879\,505}{29} \approx 426\,444$ and median (the 15th value) 397 051. Removing one district lowers the mean by about 15 100 (3.4%) but the median by only 3 498.5 (0.9%). Conclusion: the median is resistant to extreme values; the mean is not.

Worked example 3: estimates from the grouped table

Use the grouped table of lesson 1 (classes of 100 thousand; frequencies 15, 10, 3, 1, 0, 1; midpoints 350 to 850) to estimate the mean, median and mode, in thousands.

\bar{x} \approx \frac{15(350) + 10(450) + 3(550) + 1(650) + 0(750) + 1(850)}{30} = \frac{12\,900}{30} = 430

Median: $\frac{n}{2} = 15$ , and the cumulative frequency reaches 15 exactly at the end of the first class, so

\text{Median} \approx 300 + \frac{15 - 0}{15} \times 100 = 400

Mode: modal class $300 \le x \lt 400$ ; $d_{1} = 15 - 0 = 15$ , $d_{2} = 15 - 10 = 5$ :

\text{Mode} \approx 300 + \frac{15}{15 + 5} \times 100 = 375

Compare: the exact mean (441.5) and median (400.5) are close to the estimates; the mean estimate is low by about 11.5 thousand because most districts lie in the lower half of their classes. The order mode $< \lt$ median $< \lt$ mean confirms the right skew.

Warning

Common mistakes. Dividing $\sum fx$ by the number of classes instead of by $\sum f$ . Taking the median of the frequencies instead of the values. Forgetting to put ungrouped data in order before finding the median. Using class limits instead of boundaries as $L$ in the median formula for whole-number data (for marks 40–49, $L = 39.5$ ). Quoting a grouped estimate as if it were exact.

Corbettmaths, "Estimated Mean". Follow the midpoint column method, then repeat it for the district table in worked example 3 to get 430 thousand.

✦

Practice

Try each one before opening the answers. They get harder as you go.

Find the mean, median and mode of 12, 15, 9, 15, 20, 11, 15, 13.
Goals scored in 20 matches: 0 goals: 4 matches; 1: 6; 2: 5; 3: 3; 4: 2. Find the mean, median and mode.
Daily earnings of 50 motorcycle-taxi riders (thousand Frw): $0 \le x \lt 5$ : 4; $5 \le x \lt 10$ : 11; $10 \le x \lt 15$ : 18; $15 \le x \lt 20$ : 12; $20 \le x \lt 25$ : 5 (illustrative). Estimate the mean, median and mode.
Class A (30 pupils) has a mean mark of 62; class B (20 pupils) has a mean of 70. Find the mean mark of all 50 pupils.
The mean of five numbers is 8. Four of them are 6, 9, 7 and 10. Find the fifth.
Which average would you use for: (a) the shoe size a shop should order most of; (b) the typical price of a plot of land in Kigali; (c) the total maize a cooperative's 40 farmers will harvest, from a sample of 10? Give reasons.
"The average Rwandan district has about 441 500 people, so most districts are about that size." Evaluate this statement.

Show answersHide answers

Sum $= 110$ , mean $= \frac{110}{8} = 13.75$ . In order: 9, 11, 12, 13, 15, 15, 15, 20; median $= \frac{13 + 15}{2} = 14$ . Mode $= 15$ .
$\sum fx = 0 + 6 + 10 + 9 + 8 = 33$ ; mean $= \frac{33}{20} = 1.65$ . The 10th and 11th values are 1 and 2: median $= 1.5$ . Mode $= 1$ .
Midpoints 2.5, 7.5, 12.5, 17.5, 22.5; $\sum fx = 640$ ; mean $= \frac{640}{50} = 12.8$ thousand Frw. Median: $\frac{n}{2} = 25$ , cumulative 4, 15, 33, so the class is 10–15: $10 + \frac{25 - 15}{18} \times 5 \approx 12.78$ . Mode: $d_{1} = 7$ , $d_{2} = 6$ : $10 + \frac{7}{13} \times 5 \approx 12.69$ thousand Frw.
$\frac{30 \times 62 + 20 \times 70}{50} = \frac{3260}{50} = 65.2$ . (Not $\frac{62 + 70}{2} = 66$ : the classes are different sizes.)
Total $= 5 \times 8 = 40$ ; fifth number $= 40 - (6 + 9 + 7 + 10) = 8$ .
(a) Mode: the most common size. (b) Median: land prices are skewed by a few very expensive plots. (c) Mean: multiply the sample mean by 40 to estimate the total.
Misleading. 441 500 is the mean, pulled up by a few large districts, especially Gasabo; 20 of the 30 districts are smaller, and half have under about 400 500 (the median). "Most districts" are 300 000 to 500 000, with half in 300 000–400 000. For a typical district the median is the better summary.

Quick check

Key points

$\bar{x} = \frac{\sum x}{n} = \frac{\sum fx}{\sum f}$ ; for grouped data use class midpoints, which gives an estimate.
The median is the middle value of ordered data, position $\frac{n + 1}{2}$ ; grouped: $L + \frac{\frac{n}{2} - F}{f_m} \times c$ .
The mode is the most frequent value; grouped: $L + \frac{d_1}{d_1 + d_2} \times c$ .
The median resists extreme values; the mean uses every value and is pulled towards a long tail.
For Rwanda's districts (2022): mean ≈ 441 546, median ≈ 400 550; right skew, so mode $< \lt$ median $< \lt$ mean.

Sources and credits

Data: Fifth Rwanda Population and Housing Census, 2022: Main Indicators Report, National Institute of Statistics of Rwanda (Table 1).
Photo: The horizon view of one of Kigali City chain of hills named Bumbogo, taken from Kibagabaga, Mpapierre, Wikimedia Commons, CC BY-SA 4.0.
Video: Estimated Mean, Corbettmaths, embedded from YouTube.
Written by Ganzaa: the explanations, the dot plot (drawn from NISR data), the worked examples, the practice items and the questions. Household, match, earnings and class data are illustrative. This lesson uses CC BY-SA material and is shared under CC BY-SA 4.0.

1How big is a "typical" district?

2Grouped data: estimates

How big is a "typical" district?

Grouped data: estimates