Range — Measuring the Spread of Data

Introduction

In statistics, the range is the simplest measure of spread. It tells you how far apart the data values are by subtracting the smallest value from the largest value. In symbols:

Range = Highest value − Lowest value

Why do we care about spread? Two classes can have the same average (mean or median) but be very different in how the marks are distributed. The range gives a quick first check of variability: a small range means the data are bunched together; a large range means the data are more spread out.

Consider test scores out of 50.

• Class A scores: 44, 45, 45, 46, 47 → Range = 47 − 44 = 3 (very consistent)
• Class B scores: 10, 24, 33, 45, 50 → Range = 50 − 10 = 40 (very spread out)

At GCSE level, questions about range appear in contexts such as exam marks, times, lengths, temperatures, and grouped frequency tables (where you sometimes estimate the range using class boundaries). You may be asked to:

• Calculate the range from a raw list of values.
• Compare two data sets using both an average (mean/median) and a measure of spread (range or interquartile range).
• Identify how an outlier (an unusually large or small value) affects the range.
• Estimate the range from grouped data or a stem-and-leaf/box plot.

The range is quick and useful — but it is also sensitive to outliers because it uses only the two extreme values. For a more robust view of spread, later topics introduce the interquartile range (IQR) and the standard deviation. Still, the range remains a key first step in describing data.

In this tutorial you will learn to compute the range accurately from lists, tables and plots, interpret it in context, and use it alongside an average to compare two groups fairly. We’ll also cover typical exam pitfalls (like forgetting to order the data or mixing up units) and finish with graded practice from Foundation to Higher.

Key Vocabulary

• Range — A simple measure of spread: Range = highest value − lowest value. It shows how wide the data are.
• Data set — The collection of numbers you’re analysing (e.g. 8 lap times, 12 test scores).
• Maximum (highest value) — The largest number in the data set.
• Minimum (lowest value) — The smallest number in the data set.
• Spread / Variability — How “spread out” the data are. Range is the quickest way to describe spread.
• Consistency — Small range ⇒ more consistent results; large range ⇒ less consistent.
• Outlier — A value much bigger or smaller than the rest. Outliers can make the range much larger, so the range is said to be not robust.
• Ordered data — Data written from smallest to largest (or vice versa). Ordering helps you spot the min and max quickly and avoid mistakes.
• Units — Always include units with the range (cm, kg, °C, minutes, etc.). The range is a difference and uses the same unit as the data.
• Discrete data — Whole-number counts (e.g. number of goals). Range is found the same way.
• Continuous data — Measurements that can take any value in an interval (e.g. height, time). Be careful with rounding and boundaries.
• Grouped data — Data summarised into classes (intervals), e.g. 120–129 cm, 130–139 cm. When only grouped data are given, you often estimate the range using class boundaries.
• Class interval — The width of a group in a grouped frequency table (e.g. 130–139 has width 10).
• Class boundaries — Exact edges of a class used to avoid gaps (e.g. 130–139 becomes 129.5–139.5 for continuous data). For estimates of range from grouped data, use the lowest and highest boundaries.
• Exact vs estimated range — From a raw list (or stem-and-leaf) you can calculate the exact range. From a grouped table you usually give an estimated range because you don’t know individual values inside each class.
• Box plot range — The full horizontal span from the smallest value (left whisker) to the largest value (right whisker). (Note: the interquartile range (IQR) is the box width, Q3 − Q1, and is more robust to outliers.)
• Median / Mean (context) — Averages that describe centre. In comparison questions, pair an average with a measure of spread (range or IQR) to comment fully.

Core Ideas

The range is a quick measure of spread, but to use it confidently in GCSE exams you need to understand how it behaves and where it can (and cannot) be trusted. Here are the essential concepts:

• Definition: \[ \text{Range} = \text{Maximum value} - \text{Minimum value} \] This single subtraction summarises the entire spread of the data.
• Range shows consistency: - Small range → data values are close together → consistent performance. - Large range → data values are spread out → less consistent performance.
• Effect of outliers: Because the range uses only the highest and lowest values, one extreme score can change it dramatically. Example: Data = 4, 5, 6, 7 → Range = 3. If an outlier 50 is added → Range = 50 − 4 = 46.
• Comparisons: When comparing two data sets, examiners expect you to mention both a measure of average (mean or median) and a measure of spread (range or IQR). Example: “Class A has a higher median mark but also a larger range, so their performance is less consistent.”
• Exact vs estimated range: - From a raw list (or stem-and-leaf diagram), the range is exact. - From a grouped frequency table, you usually estimate by using the lowest and highest class boundaries.
• Box plots: The range is the distance from the smallest value (left whisker) to the largest value (right whisker). Exams often ask: “Use the box plots to compare medians and ranges.”
• Units: Always include the correct unit in your answer. If the data are in kilograms, the range must be given in kilograms.
• Link with averages: The range never tells you the “typical” value, only the spread. Pair it with an average to give a complete picture of the data set.
• Interpretation in context: Always phrase your conclusion in the context of the problem. Example: “The times for Runner A have a smaller range, so Runner A is more consistent than Runner B.”

Step-by-Step Method

To calculate the range correctly in GCSE questions, follow this structured approach. These steps apply whether you are working with a raw list, a table, or a graph/plot.

1. Write down the data clearly. If given a list of numbers, copy them down neatly. For tables or diagrams, note the relevant values.
2. Order the data (if needed). Rearranging the data from smallest to largest makes it easy to spot the minimum and maximum. Example: Raw data = 12, 8, 15, 9 → Ordered = 8, 9, 12, 15.
3. Identify the minimum (lowest value). Check carefully—sometimes students mistake the second-lowest value as the minimum.
4. Identify the maximum (highest value). Again, be careful not to confuse the maximum with the second-highest. On graphs, check the axis labels.
5. Subtract: Maximum − Minimum. This is the definition of range. Do the subtraction carefully, especially when dealing with decimals or negative values.
   Example: Max = 15, Min = 8 → Range = 15 − 8 = 7.
6. Include the correct unit. - If values are in cm, write “7 cm”. - If times are in minutes, write “12 minutes”. Examiners award accuracy marks for correct units.
7. Check for outliers. If one extreme value looks unusual, note how it changes the range. Exams may ask you to comment on this effect.
8. For grouped data: Use the lowest and highest class boundaries. Example: Heights grouped 140–149 cm up to 180–189 cm → Range ≈ 189.5 − 139.5 = 50 cm.
9. For box plots: Read the smallest value (left whisker) and the largest value (right whisker), then subtract.
10. Interpret in context. Always give meaning to your result. Example: “The range of 12 minutes means the slowest runner took 12 minutes longer than the fastest runner.”

Worked Examples — Foundation

Let’s practise finding the range with straightforward Foundation-tier questions. Notice how each example follows the same step-by-step method.

Example 1 — Simple raw data

The ages of 6 children are: 7, 8, 10, 9, 11, 9.

1. Order the data: 7, 8, 9, 9, 10, 11.
2. Lowest = 7, Highest = 11.
3. Range = 11 − 7 = 4.

Answer: 4 years.

Example 2 — Test scores

The marks in a quiz are: 12, 18, 15, 20, 10.

1. Order: 10, 12, 15, 18, 20.
2. Lowest = 10, Highest = 20.
3. Range = 20 − 10 = 10.

Answer: 10 marks.

Example 3 — Heights (with units)

The heights (cm) of 5 plants are: 34, 42, 39, 37, 41.

1. Order: 34, 37, 39, 41, 42.
2. Lowest = 34 cm, Highest = 42 cm.
3. Range = 42 − 34 = 8 cm.

Answer: 8 cm.

Example 4 — Times

Five friends record their running times (minutes): 12, 15, 14, 11, 17.

1. Order: 11, 12, 14, 15, 17.
2. Lowest = 11 min, Highest = 17 min.
3. Range = 17 − 11 = 6 min.

Answer: 6 minutes.

Example 5 — Negative values

Temperatures (°C) over 5 days: −3, 0, 5, −1, 2.

1. Order: −3, −1, 0, 2, 5.
2. Lowest = −3, Highest = 5.
3. Range = 5 − (−3) = 5 + 3 = 8.

Answer: 8 °C.

Example 6 — Grouped data (estimate)

A grouped table shows shoe sizes of students:

- Lowest class = 3–4 → boundary = 2.5. - Highest class = 9–10 → boundary = 10.5. - Estimated range = 10.5 − 2.5 = 8.

Answer: About 8 shoe sizes.

Worked Examples — Higher

Now let’s look at exam-style Higher-tier problems. These examples often involve larger data sets, grouped tables, box plots, or contextual interpretation. The process is the same: maximum − minimum.

Example 1 — Larger raw data set

A class records the number of text messages sent in a day: 12, 14, 22, 19, 31, 18, 25, 27, 29, 16.

1. Order: 12, 14, 16, 18, 19, 22, 25, 27, 29, 31.
2. Lowest = 12, Highest = 31.
3. Range = 31 − 12 = 19.

Answer: 19 messages.

Example 2 — Decimals

The weights (kg) of 6 boxes are: 4.8, 5.2, 4.9, 5.6, 5.1, 5.3.

1. Order: 4.8, 4.9, 5.1, 5.2, 5.3, 5.6.
2. Lowest = 4.8 kg, Highest = 5.6 kg.
3. Range = 5.6 − 4.8 = 0.8 kg.

Answer: 0.8 kg.

Example 3 — Negative and positive values

Temperatures (°C) over a week: −5, 2, 0, −1, 4, −3, 1.

1. Order: −5, −3, −1, 0, 1, 2, 4.
2. Lowest = −5, Highest = 4.
3. Range = 4 − (−5) = 9.

Answer: 9 °C.

Example 4 — Grouped frequency table

Speeds of cars (mph):

- Lowest boundary = 29.5 - Highest boundary = 69.5 - Estimated range = 69.5 − 29.5 = 40.

Answer: Approximately 40 mph.

Example 5 — Box plot

The box plot below summarises a set of exam marks:

• Lowest = 22
• Q1 = 35
• Median = 48
• Q3 = 62
• Highest = 78

Range = 78 − 22 = 56.

Answer: 56 marks.

Example 6 — Comparing two sets

Class A exam marks: Range = 12. Class B exam marks: Range = 28.

Interpretation: Class A’s results are more consistent. Class B’s results are more spread out, meaning some students did very well and others poorly.

Common Mistakes & Fixes

Although calculating the range looks simple, many students lose easy marks through avoidable slips. Here are the most frequent mistakes and how to correct them.

• Forgetting to order the data Mistake: Picking the first and last numbers from the list without checking if they are really min and max. Fix: Always write the data in order first — smallest to largest.
• Subtracting the wrong way round Mistake: Doing “lowest − highest” and getting a negative range. Fix: Always do “highest − lowest”. The range is a positive difference.
• Leaving out units Mistake: Writing “Range = 8” instead of “Range = 8 cm”. Fix: Copy the same unit from the data values into your answer.
• Using frequencies instead of data values Mistake: In a table, subtracting frequencies (counts) instead of the actual class boundaries or values. Fix: The range is about the values (heights, times, scores), not how many there are.
• Mixing up class boundaries Mistake: Using 130 and 180 instead of 129.5 and 189.5 for continuous data. Fix: Always use boundaries when estimating the range from grouped data.
• Forgetting to include both ends of a box plot Mistake: Subtracting Q3 − Q1 (that’s the IQR, not the range). Fix: For range, use maximum − minimum. For IQR, use Q3 − Q1. Don’t confuse the two.
• Ignoring outliers Mistake: Saying “Range = 60” without commenting that one extreme value (e.g. 100) caused it. Fix: If an outlier makes the range misleading, mention it in your interpretation.
• Not interpreting in context Mistake: Just giving a number without explanation. Fix: Add meaning: “The times varied by 8 minutes between fastest and slowest runner.”

Practice Questions — Foundation

Try these problems to practise calculating the range from simple data sets. Show all working and remember to include units in your answers.

1. The shoe sizes of 5 friends are: 4, 6, 5, 7, 4. Find the range.
2. The number of pets owned by 6 families: 2, 0, 3, 1, 4, 2. Work out the range.
3. The heights (cm) of 7 seedlings: 12, 15, 18, 20, 14, 16, 19. Calculate the range in cm.
4. A group of students record the times (minutes) they take to walk to school: 11, 18, 14, 16, 20. What is the range?
5. Temperatures (°C) for 5 days: −1, 3, 5, 2, −2. Find the range.
6. A table shows ages of children in a club:

   Age (years)	Frequency
   5–6	4
   7–8	6
   9–10	3

   Estimate the range.
7. A box plot shows the marks in a test: Lowest = 15, Highest = 42. Work out the range.

Practice Questions — Higher

These Higher-tier questions involve decimals, negatives, grouped data, and box plots. Use the same “maximum − minimum” method, but take care with boundaries and units.

1. The weights (kg) of 8 parcels are: 3.2, 2.8, 4.0, 3.5, 3.7, 2.9, 4.1, 3.3. Work out the range.
2. Daily temperatures (°C) for a week: −6, −2, 0, 4, 1, −3, 5. Find the range.
3. A grouped frequency table shows journey times to school:

   Time (minutes)	Frequency
   0–9	3
   10–19	7
   20–29	12
   30–39	5

   Estimate the range.
4. The box plot of marks in a maths test shows: Lowest = 22, Q1 = 34, Median = 48, Q3 = 61, Highest = 76. Find the range.
5. Compare the consistency of two football players’ goal tallies: - Player A scored goals with range = 4. - Player B scored goals with range = 11. Which player was more consistent? Explain briefly.
6. A second class has marks with a median of 55 and a range of 20. The first class has a median of 60 and a range of 40. Compare the two classes in terms of average performance and consistency.
7. The speeds (mph) of cars are grouped as follows:

   Speed (mph)	Frequency
   30–39	6
   40–49	10
   50–59	14
   60–69	5

   Estimate the range using class boundaries.

Challenge Questions

These tougher questions combine the range with other skills such as interpreting context, comparing two data sets, or working from grouped data and box plots. They reflect multi-step reasoning often seen in Higher-tier GCSE exam papers.

1. A factory measures the length (cm) of 12 rods: 98, 102, 101, 99, 100, 97, 105, 103, 99, 100, 98, 104.
   (a) Find the range. (b) Comment on whether the rods are made consistently to the required length.
2. Two football teams record goals scored in 10 matches. - Team A: mean = 2.8 goals, range = 3. - Team B: mean = 2.8 goals, range = 7. Which team is more consistent? Explain using both mean and range.
3. A grouped frequency table shows the times (minutes) taken to run a race:

   Time (minutes)	Frequency
   30–34	6
   35–39	10
   40–44	9
   45–49	5

   (a) Estimate the range. (b) Why is this only an estimate?
4. The box plots show test scores for two schools. - School X: lowest = 15, median = 48, highest = 76. - School Y: lowest = 20, median = 45, highest = 55. Compare the two schools’ performance in terms of typical score and consistency.
5. A climber records temperatures (°C) at different altitudes: −15, −12, −8, −3, 0, 2, 5, 9. (a) Find the range. (b) Explain what the range tells us about the change in conditions.
6. A survey records shoe sizes of students. Data are grouped as:

   Shoe size	Frequency
   2–3	4
   4–5	6
   6–7	8
   8–9	2

   (a) Estimate the range. (b) Suggest one reason why the true range might be smaller than your estimate.
7. Two box plots are given: - Class A: lowest = 12, Q1 = 24, median = 32, Q3 = 41, highest = 55. - Class B: lowest = 18, Q1 = 25, median = 36, Q3 = 40, highest = 42. (a) Find the range for each class. (b) Comment on which class is more consistent.

Quick Revision Sheet

Here’s a one-page summary of everything you need to know about the range. Perfect for last-minute revision before an exam.

Key Formula

Range = Maximum value − Minimum value

How to Calculate

1. Order the data (smallest to largest).
2. Pick out the lowest and highest values.
3. Subtract: highest − lowest.
4. Write the answer with correct units.

When to Use

• Describing spread (how consistent the data are).
• Comparing two sets of data (must mention both average and range).
• Quick estimate of variability when exact calculations are not needed.

Units

• Always keep the same units as the data.
• If data are in cm, range must be in cm.
• If times are in minutes, range must be in minutes.

Special Cases

• Negative values: Subtract carefully (e.g. 5 − (−3) = 8).
• Grouped data: Use class boundaries for lowest and highest groups (estimated range).
• Box plots: Use smallest (left whisker) and largest (right whisker).

Common Traps

• Subtracting the wrong way round (getting negative answers).
• Forgetting to order the data before picking extremes.
• Leaving out units (marks lost!).
• Confusing range with interquartile range.

Interpretation Phrases

• “The range of 12 marks shows the scores are spread over 12 marks.”
• “Class A has a smaller range, so their results are more consistent.”
• “The large range suggests big differences in performance.”

Quick Practice

1. Data = 4, 7, 9, 11 → Range = ? (Answer: 7)
2. Data = −2, 3, 5, 8 → Range = ? (Answer: 10)
3. Grouped data 20–29, …, 50–59 → Estimated range = 59.5 − 19.5 = ? (Answer: 40)

Answers

Practice — Foundation

1. Shoe sizes 4, 6, 5, 7, 4 → min 4, max 7 → range = 3 (sizes).
2. Pets 2, 0, 3, 1, 4, 2 → min 0, max 4 → range = 4.
3. Heights 12, 15, 18, 20, 14, 16, 19 cm → min 12, max 20 → range = 8 cm.
4. Walk times 11, 18, 14, 16, 20 min → min 11, max 20 → range = 9 minutes.
5. Temps −1, 3, 5, 2, −2 °C → min −2, max 5 → range = 7 °C.
6. Grouped ages 5–6, 7–8, 9–10 → boundaries 4.5 and 10.5 → estimated range = 6 years.
7. Box plot lowest 15, highest 42 → range = 27 (marks).

Practice — Higher

1. Parcels (kg) → min 2.8, max 4.1 → range = 1.3 kg.
2. Temps −6, −2, 0, 4, 1, −3, 5 °C → min −6, max 5 → range = 11 °C.
3. Journey times 0–9,…,30–39 → boundaries −0.5 and 39.5 → estimated range = 40 minutes.
4. Box plot lowest 22, highest 76 → range = 54 (marks).
5. Consistency: Player A (range 4) is more consistent than Player B (range 11).
6. Class 1 median 60, range 40; Class 2 median 55, range 20 → Class 1 higher typical score, Class 2 more consistent.
7. Speeds 30–39,…,60–69 → boundaries 29.5 and 69.5 → estimated range = 40 mph.

Challenge

1. Rods: min 97, max 105 → range = 8 cm. Comment: small range ⇒ lengths are fairly consistent around the target.
2. Teams: same mean (2.8) but ranges 3 vs 7 → Team A more consistent; Team B more variable.
3. Race times 30–34,…,45–49 → boundaries 29.5 and 49.5 → estimated range = 20 minutes. Reason: grouped data only; true min/max inside classes unknown.
4. School X: lowest 15, median 48, highest 76 → range = 61. School Y: lowest 20, median 45, highest 55 → range = 35. Comparison: X has higher typical score (median) but less consistent; Y is more consistent but lower typical score.
5. Climber temps −15,…,9 → min −15, max 9 → range = 24 °C. Meaning: large variation in conditions with altitude/time.
6. Shoe sizes 2–3,…,8–9 → boundaries 1.5 and 9.5 → estimated range = 8. True range may be smaller if extremes (1.5 or 9.5) don’t actually occur.
7. Box plots: Class A range = 55 − 12 = 43. Class B range = 42 − 18 = 24. More consistent: Class B (smaller range).

Conclusion & Next Steps

The range is the fastest way to describe spread: maximum − minimum. It’s perfect for quick comparisons and for pairing with an average (mean/median) to give a complete description. Remember its weakness: it depends only on the two extreme values, so outliers can distort it. In later statistics topics you’ll use more robust measures like the interquartile range (IQR) and more detailed measures like standard deviation.

• Exam habit: Always state units and interpret the range in context (consistency/reliability).
• Comparison habit: Mention both centre (mean/median) and spread (range/IQR) for full marks.
• Grouped data: Use class boundaries for an estimated range and say it’s an estimate.

Up next in the Statistics series, we’ll build on spread with Interquartile Range (IQR) and then practise comparing data with box plots.