Descriptive Statistics

Lessons

Senior 4 · Basic Mathematics

Descriptive Statistics

5 lessons

Lesson 4 of 5

Quartiles, variance and standard deviation

25 min

Same average, different story

Two mills sell maize flour in 5 kg bags. An inspector weighs 8 bags from each. Mill A: 4.98, 5.02, 5.00, 4.97, 5.03, 5.01, 4.99, 5.00 kg. Mill B: 4.80, 5.20, 5.05, 4.90, 5.15, 4.85, 5.10, 4.95 kg. (Illustrative data.) Both means are exactly 5.00 kg, yet customers of Mill B complain that some bags are 200 g short. An average alone cannot show this. We also need a measure of spread (dispersion): how far the values lie from the centre and from each other.

Definition

Range $=$ largest value $-$ smallest value. Quartiles $Q_{1}, Q_{2}, Q_{3}$ divide ordered data into four equal parts; $Q_{2}$ is the median. Interquartile range $\text{IQR} = Q_3 - Q_1$ , the spread of the middle half; quartile deviation $= \frac{Q_3 - Q_1}{2}$ .

Variance $\sigma^2$ : the mean of the squared deviations from the mean. Standard deviation $\sigma = \sqrt{\sigma^2}$ , in the same units as the data:

\sigma^2 = \frac{\sum (x - \bar{x})^2}{n} = \frac{\sum x^2}{n} - \bar{x}^2, \qquad \sigma^2 = \frac{\sum f (x - \bar{x})^2}{\sum f} = \frac{\sum f x^2}{\sum f} - \bar{x}^2 .

Finding quartiles

Ungrouped data (in order): $Q_{1}$ is at position $\frac{n + 1}{4}$ and $Q_{3}$ at position $\frac{3(n + 1)}{4}$ ; when a position is not a whole number, interpolate between the two neighbouring values. (Calculators and software sometimes use slightly different rules, which give slightly different answers.)

Grouped data: as for the median, interpolate inside the class that contains the $\frac{n}{4}$ th or $\frac{3n}{4}$ th value:

Q_1 \approx L + \frac{\frac{n}{4} - F}{f} \times c, \qquad Q_3 \approx L + \frac{\frac{3n}{4} - F}{f} \times c .

Graphically, draw the cumulative frequency curve (ogive), plotting each cumulative frequency at its class's upper boundary, and read across from $\frac{n}{4}$ , $\frac{n}{2}$ and $\frac{3n}{4}$ .

The five-number summary and the box plot

Minimum, $Q_{1}$ , median, $Q_{3}$ and maximum summarise a distribution; a box plot draws them. A value more than $1.5 \times \text{IQR}$ beyond a quartile is usually treated as an outlier and plotted on its own.

The box (the middle half of the districts) is narrow and sits to the left; the right whisker is long and Gasabo lies beyond the fence. Box plots show skew and outliers at a glance and make it easy to compare groups side by side.

Tip

Pair the measures. Report the median with the IQR (both resist outliers), and the mean with the standard deviation (both use every value). For skewed data with outliers, such as district populations or incomes, the median and IQR describe a typical member better.

✦

Worked example 1: the two mills

Find the range and the standard deviation for each mill (both means are 5.00 kg).

Mill B: $x$	4.80	5.20	5.05	4.90	5.15	4.85	5.10	4.95	Sum
$x - \bar{x}$	−0.20	0.20	0.05	−0.10	0.15	−0.15	0.10	−0.05	0
$(x - \bar{x})^2$	0.0400	0.0400	0.0025	0.0100	0.0225	0.0225	0.0100	0.0025	0.1500

\sigma_B^2 = \frac{0.15}{8} = 0.01875, \qquad \sigma_B = \sqrt{0.01875} \approx 0.137\ \text{kg}

The same working for Mill A gives $\sum (x - \bar{x})^2 = 0.0028$ , $\sigma_A^2 = 0.00035$ and $\sigma_A \approx 0.019$ kg. Ranges: A $= 5.03 - 4.97 = 0.06$ kg; B $= 5.20 - 4.80 = 0.40$ kg.

Conclusion: Mill B's bags vary about seven times as much ( $\frac{0.137}{0.019} \approx 7$ ). The deviations always add up to 0, which is why we square them before averaging.

Worked example 2: quartiles and outliers in the census data

For the 30 district populations (in order), $Q_{1}$ is at position $\frac{31}{4} = 7.75$ and $Q_{3}$ at $\frac{93}{4} = 23.25$ .

Q_1 = 365\,718 + 0.75(367\,955 - 365\,718) \approx 367\,396, \qquad Q_3 = 484\,953 + 0.25(485\,529 - 484\,953) = 485\,097

\text{IQR} = 485\,097 - 367\,396 \approx 117\,701; \qquad \text{upper fence} = Q_3 + 1.5 \times \text{IQR} \approx 661\,649

Only Gasabo (879 505) lies above the fence, so it is the one outlier; Nyagatare (653 861) is large but inside. The lower fence, $Q_1 - 1.5 \times \text{IQR} \approx 190\,844$ , is below every district.

Worked example 3: standard deviation and quartiles from the grouped table

Using midpoints $x$ (thousands) and $f$ = 15, 10, 3, 1, 0, 1, with $\bar{x} = 430$ from lesson 3:

\sum f x^2 = 15(350^2) + 10(450^2) + 3(550^2) + 1(650^2) + 0 + 1(850^2) = 5\,915\,000

\sigma^2 = \frac{5\,915\,000}{30} - 430^2 = 197\,166.7 - 184\,900 = 12\,266.7, \qquad \sigma \approx 110.8\ \text{thousand}

Quartiles: $\frac{n}{4} = 7.5$ lies in the first class, $Q_1 \approx 300 + \frac{7.5 - 0}{15} \times 100 = 350$ ; $\frac{3n}{4} = 22.5$ lies in $400 \le x \lt 500$ , $Q_3 \approx 400 + \frac{22.5 - 15}{10} \times 100 = 475$ . IQR $\approx 125$ thousand.

Compare with the exact values from the 30 numbers: $\sigma \approx 112.1$ thousand and IQR $\approx 117.7$ thousand. Grouping loses a little accuracy but keeps the picture.

Warning

Common mistakes. Forgetting the square root (variance is in "thousands squared"; standard deviation is in thousands). Using $\frac{\sum f x^2}{\sum f} - \bar{x}$ instead of $- \bar{x}^2$ . Rounding $\bar{x}$ too early in $\frac{\sum x^2}{n} - \bar{x}^2$ , which can make a small variance badly wrong. Mixing the sample formula (dividing by $n - 1$ , used to estimate a population from a sample) with the formula used here (dividing by $n$ ): state which you use.

Corbettmaths, "Medians, quartiles from histograms and grouped data". Watch the interpolation method, then check the grouped quartiles 350 and 475 of worked example 3.

✦

Practice

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

For 7, 3, 9, 12, 5, 8, 10, 4, 6, 11, 14, find the range, $Q_{1}$ , the median, $Q_{3}$ and the IQR.
Find the variance and standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.
The eggs laid per day by a hen were recorded on 20 days: 0 eggs on 2 days, 1 on 5 days, 2 on 8 days, 3 on 5 days (illustrative). Find the mean and standard deviation.
For the motorcycle-taxi earnings of lesson 3 (classes of 5 thousand Frw from 0 to 25; $f$ = 4, 11, 18, 12, 5; mean 12.8), estimate the standard deviation, $Q_{1}$ , $Q_{3}$ and the IQR.
A data set has $Q_{1} = 20$ and $Q_{3} = 32$ . Is a value of 55 an outlier? Is 3?
Every worker in a workshop gets a raise of 5 000 Frw a month. What happens to the mean wage, the range and the standard deviation? What if instead every wage is increased by 10%?
A buyer must choose between the two mills of this lesson at the same price. Use the statistics to advise her, and say what else she should check.

Show answersHide answers

In order: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14 ( $n = 11$ ). Range $= 11$ ; $Q_{1}$ (3rd value) $= 5$ ; median (6th) $= 8$ ; $Q_{3}$ (9th) $= 11$ ; IQR $= 6$ .
Mean $= 5$ ; squared deviations 9, 1, 1, 1, 0, 0, 4, 16, sum 32; $\sigma^2 = \frac{32}{8} = 4$ ; $\sigma = 2$ .
$\sum fx = 36$ , mean $= 1.8$ ; $\sum fx^2 = 0 + 5 + 32 + 45 = 82$ ; $\sigma^2 = \frac{82}{20} - 1.8^2 = 4.1 - 3.24 = 0.86$ ; $\sigma \approx 0.93$ eggs.
$\sum fx^2 = 9662.5$ ; $\sigma^2 = \frac{9662.5}{50} - 12.8^2 = 193.25 - 163.84 = 29.41$ ; $\sigma \approx 5.42$ thousand Frw. $Q_{1}$ : $\frac{n}{4} = 12.5$ , class 5–10: $5 + \frac{12.5 - 4}{11} \times 5 \approx 8.86$ . $Q_{3}$ : $37.5$ , class 15–20: $15 + \frac{37.5 - 33}{12} \times 5 \approx 16.88$ . IQR $\approx 8.02$ thousand Frw.
IQR $= 12$ ; fences $20 - 18 = 2$ and $32 + 18 = 50$ . 55 is an outlier; 3 is not (it is above 2).
Adding 5 000: the mean rises by 5 000; the range and standard deviation do not change (every value moves the same amount). Increasing by 10%: the mean, range and standard deviation are all multiplied by 1.1.
Both average 5.00 kg, but Mill A's standard deviation (0.019 kg) is far smaller than Mill B's (0.137 kg): A's bags are reliably close to 5 kg, while some of B's are 200 g short. Choose A. She should also check a larger sample, the flour quality and whether the scales were calibrated.

Quick check

Key points

Spread matters as much as the average: two sets with the same mean can differ greatly.
Quartiles: ungrouped at positions $\frac{n + 1}{4}$ and $\frac{3(n + 1)}{4}$ ; grouped by interpolation or from the ogive. IQR $= Q_{3} - Q_{1}$ .
Outliers lie more than $1.5 \times \text{IQR}$ beyond a quartile; a box plot shows them with the five-number summary.
$\sigma^2 = \frac{\sum f x^2}{\sum f} - \bar{x}^2$ and $\sigma = \sqrt{\sigma^2}$ , in the units of the data.
Adding a constant leaves the spread unchanged; multiplying by $k$ multiplies the standard deviation by $k$ .

Sources and credits

Data: Fifth Rwanda Population and Housing Census, 2022: Main Indicators Report, National Institute of Statistics of Rwanda (Table 1).
Video: Medians, quartiles from histograms and grouped data, Corbettmaths, embedded from YouTube.
Written by Ganzaa: the explanations, the ogive and box plot (drawn from NISR data), the worked examples, the practice items and the questions. The mill, hen and wage data are illustrative.

1Same average, different story

2Finding quartiles

3The five-number summary and the box plot

Same average, different story

Finding quartiles

The five-number summary and the box plot