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Descriptive statistics

Descriptive statistics summarizes a data set with a few numbers and charts: where the center is, how spread out the data are, whether they are symmetric and whether there are unusual values. This topic explains the mean, median and mode, variance and standard deviation (sample and population), quartiles, the box plot and outliers.

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Central tendency

The mean $\overset{x}{ˉ} = \frac{\sum x _{i}}{n}$ is the balance point of the data. The median is the middle value with the data sorted (with even $n$ , the average of the two central ones). The mode is the most repeated value. The mean uses every value, but a single extreme value drags it; the median does not notice.

A value of 48 among values from 12 to 16 raises the mean to 17.5; the median stays at 14.5.

Spread

The variance is the average of the squared deviations from the mean, and the standard deviation its square root (in the same units as the data):

s^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1} σ^{2} = \frac{\sum ( x _{i} - μ ) ^{2}}{N}

Divide by $n - 1$ when the data are a sample of something larger (the sample mean already «used up» one degree of freedom and dividing by $n$ would underestimate the variance), and by $N$ when they are the whole population. The coefficient of variation $s /∣ \overset{x}{ˉ} ∣$ compares the spread of data with different means.

Quartiles, box plot and outliers

The quartiles $Q_{1}$ , $Q_{2}$ (the median) and $Q_{3}$ leave 25 %, 50 % and 75 % of the data below them; the interquartile range $I QR = Q_{3} - Q_{1}$ measures the spread of the central 50 %. There are several ways to interpolate them; the tool uses Excel's PERCENTILE.INC: the position of percentile $p$ is $1 + p (n - 1)$ in the sorted data.

A value is an outlier if it falls outside Tukey's fences, $Q_{1} - 1.5 I QR$ and $Q_{3} + 1.5 I QR$ . The box plot sums it all up: the box goes from $Q_{1}$ to $Q_{3}$ , the line is the median, the whiskers reach the last value within the fences and outliers are drawn apart.

Box plot of the data 2, 4, 4, 4, 5, 5, 7, 9 (example 1).

Skewness tells which side the tail of the distribution is on (positive: tail to the right, mean greater than the median) and kurtosis whether its tails are heavier than those of a normal distribution.

The tool computes all these measures with the same definitions as Excel (quartiles, adjusted skewness and kurtosis), draws the histogram or frequency bars, the box plot and the frequency table, and shows the steps of each calculation.

Common mistakes

Computing the median without sorting the data.
Mixing $n$ and $n - 1$ : with a sample, the standard deviation divides by $n - 1$ .
With a frequency table, forgetting that $n$ is the sum of the frequencies, not the number of distinct values.
Deleting an outlier just like that: it may be a measurement error or the most important value; it must be investigated.

Worked examples

Example 1All the measures by hand

Compute the mean, median, mode, standard deviation (sample and population), quartiles and outliers of 2, 4, 4, 4, 5, 5, 7, 9.

Solution

$n = 8$ , $\sum x = 40$ , $\overset{x}{ˉ} = 5$ . Sorted, the middle ones are 4 and 5: median $4.5$ . Mode: 4 (three times).
Squared deviations: $9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32$ . $s^{2} = 32/7 = 4.57143$ , $s = 2.13809$ ; $σ^{2} = 32/8 = 4$ , $σ = 2$ .
$Q_{1}$ : position $1 + 0.25 \cdot 7 = 2.75$ , between the 2nd and 3rd values (4 and 4) → $4$ . $Q_{3}$ : position $6.25$ , between 5 and 7 → $5 + 0.25 \cdot 2 = 5.5$ . $I QR = 1.5$ .
Fences: $4 - 2.25 = 1.75$ and $5.5 + 2.25 = 7.75$ . The 9 is outside: it is an outlier.

\overset{x}{ˉ} = 5

, median

4.5

, mode

4

s = 2.13809

σ = 2

Q_{1} = 4

Q_{3} = 5.5

; outlier: 9.

Solve it in Simuveliq

In Math → Statistics, type or paste in Data 2 4 4 4 5 5 7 9, with «The data are: a sample (n − 1)». The cards give Mean 5, Median 4.5, Mode 4 (3×), Sample standard deviation 2.13809, population 2, Q₁ 4, Q₃ 5.5, Tukey fences 1.75 · 7.75 and Outliers 9.

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Example 2Exam scores

Summarize the 15 scores 7.5, 8, 6, 9, 5.5, 7, 8.5, 6, 7, 10, 4, 7, 8, 6.5, 9.

Solution

$\sum x = 109$ , $\overset{x}{ˉ} = 109/15 = 7.26667$ . Sorted, the 8th is 7: median 7, which is also the mode (three times).
$s = 1.55686$ and $C V = 1.55686/7.26667 = 21.42 %$ : moderate spread.
$Q_{1} = 6.25$ , $Q_{3} = 8.25$ , $I QR = 2$ ; fences $3.25$ and $11.25$ : no outliers (the 4 is inside).

\overset{x}{ˉ} = 7.26667

, median 7,

s = 1.55686

C V = 21.42 %

, no outliers.

Solve it in Simuveliq

In Math → Statistics, type or paste in Data the scores (it is the first example in the list), with «The data are: a sample (n − 1)». Besides the summary, it draws the histogram with 5 classes (Sturges' rule) and the box plot.

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Example 3Data with frequencies

In a survey, 6 families have 0 children, 14 have 1, 19 have 2, 9 have 3, 3 have 4 and 1 has 5. Find the mean, the median and the standard deviation.

Solution

$n = 6 + 14 + 19 + 9 + 3 + 1 = 52$ families. $\sum f x = 0 + 14 + 38 + 27 + 12 + 5 = 96$ , so $\overset{x}{ˉ} = 96/52 = 1.84615$ children.
Median: the 26th and 27th values; the cumulative frequencies are 6, 20, 39…, so both are 2. Mode: 2.
$s = \frac{\sum f ( x - x ˉ ) ^{2}}{n - 1} = 1.1442$ .

n = 52

\overset{x}{ˉ} = 1.84615

, median 2, mode 2,

s = 1.1442

Solve it in Simuveliq

In Math → Statistics, type or paste in Data 0 1 2 3 4 5 and in Frequencies 6 14 19 9 3 1, with «The data are: a sample (n − 1)». It gives n 52, Mean 1.84615, Median 2, Mode 2 (19×) and sample deviation 1.1442, with the table of relative and cumulative frequencies.

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Example 4An outlier

Ten measurements give 12, 15, 14, 13, 16, 15, 14, 13, 15 and 48. Which measure of center describes them best? Are there outliers?

Solution

$\overset{x}{ˉ} = 175/10 = 17.5$ , larger than nine of the ten values. Median: the 5th and 6th sorted values, 14 and 15 → $14.5$ .
$Q_{1} = 13.25$ , $Q_{3} = 15$ , $I QR = 1.75$ ; fences $13.25 - 2.625 = 10.625$ and $15 + 2.625 = 17.625$ . The 48 is an outlier.
The median (14.5) describes the group better; the high skewness (3.089) and kurtosis (9.665) give away the unusual value.

Median 14.5 (the mean, 17.5, is inflated); outlier: 48.

Solve it in Simuveliq

In Math → Statistics, type or paste in Data the ten values (it is the «With an outlier» example), with «The data are: a sample (n − 1)». It gives Mean 17.5, Median 14.5, Tukey fences 10.625 · 17.625 and Outliers 48, which the box plot draws with a circle.

Open in the simulator →

Descriptive statistics

Central tendency

Spread

Quartiles, box plot and outliers

Common mistakes

Worked examples

Example 1All the measures by hand

Solution

Solve it in Simuveliq

Example 2Exam scores

Solution

Solve it in Simuveliq

Example 3Data with frequencies

Solution

Solve it in Simuveliq

Example 4An outlier

Solution

Solve it in Simuveliq

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