How to Solve: Comparing Distributions and Drawing Conclusions

How to Solve: Comparing Distributions and Drawing Conclusions

GCSE & A-Level Maths

Introduction

"This skill turns raw data into full marks—examiners love questions where you compare two distributions and write a conclusion. On GCSE Paper 2 or A-Level Paper 3, this could be worth 6–8 marks, and it’s one of the easiest ways to pick up method marks if you follow the steps."

What You Need To Know First

1. Measures of central tendency – mean, median, mode.
2. Measures of spread – range, interquartile range (IQR), standard deviation.
3. Box plots and histograms – how to read and interpret them.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Identify the two distributions - Are they in tables, box plots, histograms, or cumulative frequency graphs? - Label them clearly (e.g., "Group A" and "Group B").

Step 2: Compare central tendency (average) - Mean or median? Use median if data is skewed or has outliers. - Write: "The median of Group A is [X], which is higher/lower than Group B’s median of [Y]."

Step 3: Compare spread (how varied the data is) - Range or IQR? Use IQR if there are outliers. - Write: "The IQR of Group A is [X], which is smaller/larger than Group B’s IQR of [Y], meaning Group A is more/less consistent."

Step 4: Check for skewness or outliers - Look at the shape of histograms or box plots. - Write: "Group A is right-skewed, while Group B is roughly symmetrical."

Step 5: Write a conclusion in context - Link back to the question (e.g., "Which group performed better?"). - Use comparative language: "Group A has a higher median and a smaller IQR, meaning they performed better and more consistently."

Worked Examples

Example 1 – Basic (Box Plots)

Question: Two classes took a maths test. Their results are shown in the box plots below. Compare the distributions.

Step 1: Label the groups (Class X and Class Y). Step 2: Compare medians. - Class X median = 65 - Class Y median = 70 → "Class Y has a higher median, meaning on average, they scored better."

Step 3: Compare IQR. - Class X IQR = 75 – 55 = 20 - Class Y IQR = 80 – 60 = 20 → "Both classes have the same IQR, meaning they have similar consistency."

Step 4: Check for skewness/outliers. - Class X has a lower whisker (55) and an outlier at 40. - Class Y is roughly symmetrical. → "Class X is left-skewed with an outlier, while Class Y is more balanced."

Step 5: Conclusion. "Class Y performed better on average (higher median) and had no outliers, while Class X had an outlier and was more spread out at the lower end."

What we did and why: - We compared medians first because they’re less affected by outliers. - We used IQR instead of range because of the outlier in Class X. - We described skewness to give a full picture.

Example 2 – Medium (Histograms)

Question: Two factories produce light bulbs. Their lifespans (in hours) are shown in the histograms below. Compare the distributions.

Step 1: Label the groups (Factory A and Factory B). Step 2: Compare medians (estimate from histograms). - Factory A median ≈ 1200 hours - Factory B median ≈ 1100 hours → "Factory A’s bulbs last longer on average."

Step 3: Compare spread. - Factory A range ≈ 1500 – 900 = 600 - Factory B range ≈ 1300 – 900 = 400 → "Factory B has a smaller range, meaning its bulbs are more consistent."

Step 4: Check skewness. - Factory A is roughly symmetrical. - Factory B is right-skewed (long tail on the right). → "Factory B has some bulbs that last much longer than the rest."

Step 5: Conclusion. "Factory A produces bulbs with a longer average lifespan (higher median), but Factory B’s bulbs are more consistent (smaller range). However, Factory B has some very long-lasting bulbs (right-skewed)."

What we did and why: - We estimated medians from histograms because exact values weren’t given. - We used range because there were no clear outliers. - We noted skewness to explain unusual patterns.

Example 3 – Exam-Style (Cumulative Frequency)

Question: The cumulative frequency graphs below show the waiting times (in minutes) for two doctors’ appointments. a) Compare the medians. b) Compare the interquartile ranges. c) Which doctor’s appointments are more consistent? Justify your answer.

Step 1: Identify the groups (Doctor P and Doctor Q). Step 2a: Find medians (50th percentile). - Doctor P median = 20 mins - Doctor Q median = 25 mins → "Doctor P has a lower median waiting time."

Step 2b: Find IQRs (Q3 – Q1). - Doctor P: Q1 = 15, Q3 = 30 → IQR = 15 - Doctor Q: Q1 = 20, Q3 = 35 → IQR = 15 → "Both doctors have the same IQR."

Step 2c: Compare consistency. - Same IQR means similar spread in the middle 50%. - But Doctor P’s range = 40 – 5 = 35 - Doctor Q’s range = 45 – 10 = 35 → "Both have the same overall spread, but Doctor P’s median is lower, meaning most patients wait less time."

Step 3: Conclusion. "Doctor P’s appointments are more consistent in terms of average wait time (lower median), but both doctors have the same variability in the middle 50% of waiting times."

What we did and why: - We used cumulative frequency to find exact quartiles. - We compared both IQR and range to give a full answer. - We linked back to the question ("more consistent").

Common Mistakes

Exam Traps

1-Minute Recap

"Right, listen up—this is your last-minute checklist for comparing distributions:

1. Label your groups (A and B, or whatever the question says).
2. Compare averages first—median if skewed, mean if not.
3. Compare spread—IQR if outliers, range if not.
4. Check the shape—is it skewed? Are there outliers?
5. Write a conclusion—link it back to the question. Use words like ‘higher,’ ‘more consistent,’ ‘skewed.’

Examiners want to see you use the data, not just guess. If you do all five steps, you’ll pick up every mark. Now go smash it!