Statistics and data analysis

Statistics is about collecting and organising data and describing it compactly with a few numbers. Exams ask about the mean, median, mode, range, variance and reading charts. Problems of the type "a new value was added — how did the mean change?" are especially common.

§1. Data sets and frequencies

• A sample is a set of observed values; their number is the sample size n.
• Frequency — how many times a value occurs; relative frequency — frequency / n (often as a percentage).
• An ordered series lists the values in increasing order.

All relative frequencies add up to 1 (or 100%) — a handy way to check your work.
For example, marks of 10 students: 5 — 5 times, 4 — 3 times, 3 — 2 times. Relative frequencies: 50%, 30%, 20%.

§2. Measures of central tendency

Each measure has its own meaning: the mean uses every value but is sensitive to extreme values ("outliers"); the median ignores them. For 3, 4, 5, 100 we have x̄ = 28 but median 4.5 — the median shows the "typical" value better.
A series may have several modes or none at all (if every value occurs once).

Worked example

For 2, 3, 3, 5, 7 find the mean, mode, median and range.

1. x̄ = (2 + 3 + 3 + 5 + 7)/5 = 20/5 = 4.
2. Mode — 3 (occurs twice).
3. The series is ordered; the middle (third) value gives median 3.
4. Range: 7 − 2 = 5.

Worked example

Find the average mark for the marks above (5 five times, 4 three times, 3 twice).

1. Weighted mean: (5·5 + 4·3 + 3·2)/10 = (25 + 12 + 6)/10.
2. x̄ = 43/10 = 4.3.

§3. The mean when data change

The key idea is to switch from the mean to the sum: sum = x̄·n. Means cannot be added or subtracted directly.

• If a is added to every value: the mean, median and mode increase by a; the range and variance stay the same.
• If every value is multiplied by k: the mean is multiplied by k, the variance by k².

Worked example

The mean height of 20 students is 140 cm. After 2 more students join, the mean becomes 141 cm. Find the total height of the two new students.

1. Old total: 20·140 = 2800.
2. New total: 22·141 = 3102.
3. 3102 − 2800 = 302 cm.

§4. Spread: variance and standard deviation

Two series with the same mean can be very different: 5, 5, 5 and 0, 5, 10. Variance measures how widely values are "scattered" around the mean:

Why squares? The deviations themselves always sum to 0 (positive and negative cancel), while squares are never negative.

Worked example

Find the variance and standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

1. x̄ = 40/8 = 5.
2. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, sum 32.
3. D = 32/8 = 4, σ = 2.

§5. Charts

• Bar chart — for comparing categories; bar height is the frequency.
• Pie chart — shows parts of a whole. The full circle is 360°:

• Line graph — shows change over time.

When reading a chart, check the units and where the axes start first: if an axis does not start at 0, differences between bars look larger than they are. In a frequency bar chart the heights of all bars add up to the sample size n.

Worked example

10 of 40 students chose football. What is the angle of this sector in a pie chart?

1. Share: 10/40 = 25%.
2. 25·3.6° = 90°.

§6. Range and outliers

The mean does not tell the whole story. For example, the data sets 5, 5, 5 and 0, 5, 10 have the same mean, 5, but the second is far more spread out. The simplest measure of spread is the range:

The range is easy to compute, but it depends on only the two extreme values. For a more careful analysis the variance and standard deviation from §4 are used.
A data set may contain a value that stands far apart from the rest — an outlier (for example, a mistyped 1700 cm among the heights of pupils in a class). Such a value affects the statistics differently:

• The mean is very sensitive to it: one large number changes the whole sum.
• The median hardly changes: it depends only on the order of the values.
• The range increases sharply, because it is calculated from the extreme value itself.

That is why for uneven data such as incomes or house prices the median is more informative than the mean. When reading a report, notice which measure is used: the ‘average salary’ may be far higher than what most people earn.

Worked example

The monthly salaries of 8 employees of a small firm (million so‘m) are 3, 3, 4, 4, 5, 5, 6, 30. Find the mean, the median and the range. What changes if the manager’s salary (30) is removed?

1. The sum is 60, the mean 60/8 = 7.5.
2. Median (n = 8 is even): (4 + 5)/2 = 4.5.
3. Range: 30 − 3 = 27.
4. Without 30: mean 30/7 ≈ 4.29, median — the 4th value, 4, range 6 − 3 = 3.
5. Conclusion: the mean fell from 7.5 to 4.29, while the median changed by only 0.5 — the median is robust to outliers.

Check yourself

1. Find the mean, median and mode of 6, 2, 8, 2, 7.

2. Find the median of 3, 9, 1, 7, 5, 11.

3. The mean of 5 numbers is 12. After one is removed, the mean of the other 4 is 10. Which number was removed?

4. Find the variance of 2, 4, 6, 8.

5. A series has mean 10 and variance 4. Each value is increased by 3. New mean and variance?

6. The pie-chart angle for 15%.

7. Find the range of 12, 7, 19, 4, 10.

8. 100 is added to the data set 2, 4, 6, 8. What are the new median and mean?

Common mistakes

• Finding the median without ordering the data.
• Taking a single middle number as the median when n is even.
• Computing a new mean from old means — work with sums instead.
• In a weighted mean, averaging only the distinct values and ignoring frequencies.
• Forgetting to divide by n or to square in the variance formula.
• Drawing conclusions from the mean alone when the data contain an outlier — compare it with the median.