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Comparing Data Sets0%

Statistics · Topic 1 of 2

Comparing Data Sets

Video lesson4 worked examples

Theory

Candidates must calculate averages (mean or median) and measures of spread (Standard Deviation or Interquartile Range) to analyse data.

Crucial Update: The National 5 syllabus now requires the calculation of the Interquartile Range (IQR) ( $Q_3 - Q_1$ ), having removed the Semi-Interquartile Range (SIQR).

Standard Deviation ( $s$ ): This measures how spread out data is around the mean. The formula is provided on the exam sheet in two forms:

s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}

and

s = \sqrt{\frac{\sum x^2 - (\sum x)^2 / n}{n - 1}}

Comparing Data

When comparing two datasets, you must provide two distinct statements using standard Qualifications Scotland phraseology:

Compare the average (e.g., "On average, [Subject A] was higher/lower...").
Compare the spread using standard deviation or IQR (e.g., "The data for [Subject A] was more/less consistent" or "more/less varied").

You must not just list the numbers; the context and comparative words are required.

The Golden Rule: a comparison always needs two statements — one about the average (which is bigger) and one about the spread (which is more consistent). A smaller standard deviation or IQR means the data is more consistent / less spread out.

⚠️ Common Examiner Traps

Divide by $n - 1$ : the standard-deviation formula uses $n - 1$ , not $n$ .
Even-sized data: with an even number of values the median is the average of the two middle numbers, and the halves for the quartiles do not include a shared middle value.
Context and comparison words: “more consistent”, “on average higher” — bare numbers with no comparison earn no marks.
Order the data first: quartiles and the median only work on data written from smallest to largest.

Worked examples

Example 1

Find the mean and standard deviation of the dataset: 2, 4, 6, 8, 10.

Step 1: Calculate the mean ( $\bar{x}$ ):

\bar{x} = \frac{2+4+6+8+10}{5} = \frac{30}{5} = 6

Step 2: Calculate $(x - \bar{x})^2$ for each value:

$(2 - 6)^2 = 16$
$(4 - 6)^2 = 4$
$(6 - 6)^2 = 0$
$(8 - 6)^2 = 4$
$(10 - 6)^2 = 16$

Step 3: Sum these values ( $\sum(x - \bar{x})^2$ ):

16 + 4 + 0 + 4 + 16 = 40

Step 4: Substitute into the formula ( $n = 5$ , so $n - 1 = 4$ ):

s = \sqrt{\frac{40}{4}} = \sqrt{10}

Answer: $s \approx 3.16$ .

Example 2

A teacher compares the test results of two classes. Class A has a mean of 65% and a standard deviation of 4.2. Class B has a mean of 72% and a standard deviation of 1.8. Make two valid comparisons.

Statement 1 (Average): On average, the test scores in Class B were higher ( $72 > 65$ ).

Statement 2 (Spread): The test scores in Class B were more consistent (or less varied) because their standard deviation is lower ( $1.8 < 4.2$ ).

Example 3

Calculate the median and Interquartile Range (IQR) for the following set of data: 12, 15, 18, 20, 22, 25, 28.

Step 1: The data is already ordered from lowest to highest.

Step 2: Find the median ( $Q_2$ ), which is the middle value:

Q_2 = 20

Step 3: Find the lower quartile ( $Q_1$ ) by finding the median of the lower half (12, 15, 18):

Q_1 = 15

Step 4: Find the upper quartile ( $Q_3$ ) by finding the median of the upper half (22, 25, 28):

Q_3 = 25

Step 5: Calculate the IQR:

\text{IQR} = Q_3 - Q_1 = 25 - 15 = 10

Answer: Median = 20, IQR = 10.

Example 4

Median & IQR of an Even-sized Set

Calculate the median and interquartile range of: 3, 5, 8, 9, 12, 14, 17, 20.

Step 1: There are 8 values (an even number), so the median is the average of the two middle values, the 4th and 5th:

Q_2 = \frac{9 + 12}{2} = 10.5

Step 2: The lower half is 3, 5, 8, 9. Its median (the lower quartile) is the average of 5 and 8:

Q_1 = \frac{5 + 8}{2} = 6.5

Step 3: The upper half is 12, 14, 17, 20. The upper quartile is the average of 14 and 17:

Q_3 = \frac{14 + 17}{2} = 15.5

Step 4: The IQR is $Q_3 - Q_1$ :

\text{IQR} = 15.5 - 6.5 = 9

Answer: median = 10.5, IQR = 9.

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10 Data Sets questions with answers and video solutions