Frequency Tables — Organising Data for Averages and Graphs

Introduction

When you collect data, it often starts as a long list of values. For example, the shoe sizes of 30 students might be written as: 5, 6, 6, 5, 7, 6, 8, 5, … and so on. Looking at this raw list makes it difficult to see patterns. That’s why statisticians use frequency tables.

A frequency table summarises data by showing each unique value (or group of values) alongside the number of times it occurs (its frequency). This makes the data clearer, easier to analyse, and ready to use in graphs or further calculations.

Frequency tables are used for both discrete data (exact counts, like number of siblings) and continuous data (measured values, like height or time, grouped into intervals).

In GCSE exams, you will be expected to:

• Construct frequency tables from raw data.
• Use them to find totals, means, medians, modes, and ranges.
• Interpret grouped frequency tables and estimate averages.
• Link tables to bar charts, histograms, and other plots.

Key Vocabulary

Before working with frequency tables, it’s important to understand the key terms you’ll see in exam questions and in real-life data handling.

• Data — The raw information collected, e.g. test scores, shoe sizes, or times.
• Frequency — The number of times a value (or group of values) occurs. Example: If “score = 7” appears 4 times, its frequency is 4.
• Tally — A quick mark system for counting data. Every group of 5 is shown as four vertical strokes with a diagonal line across. Tallies are often used when building a table.
• Discrete data — Data that take exact values (often counts). Example: Number of siblings (0, 1, 2…).
• Continuous data — Data that can take any value in a range, usually measurements. Example: Height (152.3 cm, 162.8 cm, etc.), time (12.5 minutes). These are grouped into class intervals for tables.
• Class interval — A group or range of values used to organise continuous data. Example: “10–19”, “20–29”.
• Midpoint — The middle value of a class interval, often used to estimate means. Example: Midpoint of 10–19 = 14.5.
• Cumulative frequency — A running total of frequencies, showing how many values are less than or equal to a certain amount. Often used for median and box plots.
• Relative frequency — Frequency expressed as a fraction or percentage of the total. Example: If 6 out of 30 students scored 10, relative frequency = 6 ÷ 30 = 0.2 (20%).

Core Ideas

Frequency tables simplify large sets of data and make them usable for analysis. Here are the key ideas behind constructing and using them.

• Organising data — Instead of listing every individual value, a frequency table groups identical or similar values and counts them. This shows patterns at a glance.
• Totals — The sum of frequencies always equals the total number of data points collected.
• Discrete vs Continuous — - Discrete data: frequencies are attached to individual values. - Continuous data: values are grouped into intervals, because every value is unique and data ranges are wide.
• Estimating averages — - For discrete data, you can find the exact mean, median, and mode from the table. - For grouped continuous data, you estimate the mean using midpoints, and the median/mode by interpolation.
• Link to charts — Frequency tables are the basis of bar charts, pie charts, histograms, and frequency polygons. Once data are summarised, they can be displayed visually.
• Cumulative frequency — Adding a running total column allows you to plot cumulative frequency graphs, which are used to estimate medians, quartiles, and percentiles.
• Relative frequency — By dividing each frequency by the total, you can convert a table into percentages or probabilities.

In exams, frequency tables often appear in two forms:

1. Raw tally → frequency tables: You’re asked to count and fill in the table.
2. Grouped data → estimation: You’re asked to calculate an estimated mean or comment on spread using class intervals.

Step-by-Step Method

Here’s a clear method for constructing and using frequency tables in GCSE Statistics. Follow these steps carefully depending on whether the data are discrete or continuous.

1. Building a Frequency Table (Discrete Data)

1. Write down all possible data values in a column (e.g. shoe sizes: 3, 4, 5, 6, 7, 8).
2. Go through the raw data and add a tally mark for each occurrence.
3. Total the tallies to get the frequency for each value.
4. Check that the sum of all frequencies equals the number of data points collected.

2. Building a Grouped Frequency Table (Continuous Data)

1. Decide on class intervals that cover the full range without overlap. Example: Heights (cm) grouped into 140–149, 150–159, etc.
2. Tally the raw data into the correct intervals.
3. Total the tallies to find frequencies for each class.
4. Add an extra column for the midpoint of each interval if estimating averages.

3. Estimating the Mean from a Grouped Table

1. Multiply each midpoint by its frequency (fx).
2. Find the total of all fx values.
3. Find the total of all frequencies (N).
4. Estimate the mean using: \[ \bar{x} = \frac{\sum (f \times x)}{\sum f} \]

4. Finding the Median from a Frequency Table

1. For discrete data, list cumulative frequencies until you reach the middle value.
2. For grouped data, use the cumulative frequency column and interpolation to estimate the median class.

5. Mode from a Frequency Table

1. For discrete data, the mode is simply the value with the highest frequency.
2. For grouped data, the modal class is the interval with the highest frequency.

6. Linking to Graphs

• Bar charts and pie charts come directly from discrete frequency tables.
• Histograms and frequency polygons come from grouped frequency tables.
• Cumulative frequency tables are used to plot cumulative frequency graphs for quartiles and percentiles.

Worked Examples — Foundation

Let’s look at some straightforward examples of frequency tables at the Foundation level.

Example 1 — Discrete Data (Tally & Frequency)

20 students were asked how many siblings they have. The results are: 0, 1, 2, 1, 3, 2, 2, 0, 1, 2, 3, 2, 1, 0, 4, 1, 2, 2, 3, 1.

Total frequency = 20, matching the number of students. The mode is 2 (most common number of siblings).

Example 2 — Grouped Frequency Table (Continuous Data)

The heights (cm) of 30 students were measured. The results are summarised below:

Total = 30 students. The modal class is 160–169 cm (highest frequency = 12).

Example 3 — Estimating the Mean

Using the height table above, add midpoints and calculate:

\(\sum f = 30\), \(\sum (f \times x) = 4835\). Estimated mean = \( \tfrac{4835}{30} = 161.2 \) cm.

Example 4 — Cumulative Frequency

Using the same height data, add a cumulative frequency column:

The 15th value (median position) lies in the 160–169 class. So, the median height is estimated to be within that group.

Worked Examples — Higher

At Higher level, frequency table questions often involve larger data sets, grouped intervals, and estimation of averages using more advanced techniques.

Example 1 — Estimating the Mean (Grouped Data)

The times (minutes) taken by 50 students to finish a quiz are grouped as follows:

\(\sum f = 50\), \(\sum (f \times x) = 1095\). Estimated mean = \( \tfrac{1095}{50} = 21.9 \) minutes.

Example 2 — Finding the Median with Interpolation

The grouped frequency table shows marks in a test:

Total = 50 students. Median = 25th value. This lies in the 20–29 interval. Use interpolation formula:

\[ \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times w \] where:

• \(L = 20\) (lower boundary of median class)
• \(N/2 = 25\)
• \(CF = 12\) (cumulative frequency before class)
• \(f = 13\) (frequency of class)
• \(w = 10\) (class width)

Median = \(20 + \big(\tfrac{25 - 12}{13}\big) \times 10 = 20 + (13/13 \times 10) = 30\).

Example 3 — Modal Class Estimation

From the same test score table, the highest frequency is 15 in the 30–39 class. So the modal class is 30–39 marks. In GCSE, sometimes you may be asked to apply the modal formula for a more precise estimate, but usually identifying the modal class is enough.

Example 4 — Relative Frequency

From Example 1, 18 students finished in 20–29 minutes. Relative frequency = \(18 ÷ 50 = 0.36\) (36%). This can be expressed as a probability: P(student finishes in 20–29 mins) = 0.36.

Common Mistakes & Fixes

Frequency tables may look simple, but many students lose marks through small errors. Here are the most frequent mistakes — and how to avoid them.

• Forgetting totals Mistake: Not checking that the sum of all frequencies matches the number of data points. Fix: Always add a “Total” row at the bottom and verify.
• Overlapping or inconsistent intervals Mistake: Using intervals like 0–10, 10–20 (where 10 appears in both). Fix: Write intervals consistently (e.g. 0–9, 10–19, 20–29).
• Wrong midpoints Mistake: Taking class boundaries instead of the midpoint when estimating the mean. Fix: Midpoint = (lower boundary + upper boundary) ÷ 2.
• Using frequency instead of frequency density in histograms Mistake: Treating grouped data histograms like bar charts. Fix: Calculate frequency density: \[ \text{FD} = \frac{\text{Frequency}}{\text{Class width}} \]
• Mixing up modal value and modal class Mistake: Writing a single value as the mode when the data are grouped. Fix: For grouped data, only the modal class can be found (not the exact value).
• Median position errors Mistake: Forgetting to use cumulative frequency for grouped data, or dividing by 2 incorrectly. Fix: Median = position \((N+1)/2\) for discrete, \(N/2\) with interpolation for grouped.
• Relative frequency confusion Mistake: Writing frequencies as percentages without dividing by the total. Fix: Relative frequency = frequency ÷ total. Multiply by 100 if percentage is required.

Practice Questions — Foundation

These problems will help you practise constructing and interpreting simple frequency tables.

1. The ages of 12 children are: 8, 9, 8, 10, 11, 9, 8, 12, 10, 9, 11, 8.
   Construct a frequency table showing each age and its frequency.
2. A teacher records the number of books borrowed by 15 students: 0, 1, 2, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 2, 1.
   (a) Create a frequency table. (b) State the mode.
3. Shoe sizes of 20 students are: 4, 5, 5, 6, 5, 7, 6, 4, 5, 6, 5, 7, 8, 6, 6, 5, 4, 5, 6, 7.
   (a) Make a frequency table. (b) How many students had shoe size 6?
4. A dice is rolled 30 times with results: 1, 3, 2, 6, 4, 5, 3, 2, 6, 1, 3, 5, 4, 6, 6, 2, 1, 3, 2, 5, 4, 6, 6, 2, 1, 5, 3, 4, 2, 6.
   Construct a frequency table for the six outcomes.
5. A survey of 25 students records hours spent on homework each week: 0–2 h → 4 students, 3–5 h → 8 students, 6–8 h → 9 students, 9–11 h → 4 students.
   (a) Complete the grouped frequency table. (b) State the modal class.
6. The test scores of 10 students are: 7, 9, 10, 8, 6, 10, 8, 9, 7, 6.
   (a) Draw a tally and frequency table. (b) Find the mean score.

Practice Questions — Higher

These questions involve grouped data, estimation, and interpretation. Show all working clearly, especially when calculating averages or using interpolation.

1. The table shows the times (minutes) 40 students took to finish a puzzle:

   Time (mins)	Frequency
   0–9	6
   10–19	10
   20–29	15
   30–39	9

   (a) Add a midpoint and \(f \times x\) column. (b) Estimate the mean time.
2. A frequency table of marks in a test is shown below:

   Marks	Frequency
   0–9	4
   10–19	6
   20–29	10
   30–39	15
   40–49	5

   (a) Construct a cumulative frequency table. (b) Use it to estimate the median mark.
3. The grouped frequency table shows heights of 60 plants:

   Height (cm)	Frequency
   0–9	5
   10–19	12
   20–29	18
   30–39	15
   40–49	10

   (a) Identify the modal class. (b) Estimate the mean height.
4. A set of 50 students’ shoe sizes are grouped:

   Shoe Size	Frequency
   3–4	4
   5–6	15
   7–8	20
   9–10	11

   (a) Find the cumulative frequencies. (b) Estimate the lower quartile (Q1) and upper quartile (Q3). (c) Hence find the interquartile range (IQR).
5. A relative frequency table shows how often buses arrived within different waiting times:

   Waiting time (mins)	Relative Frequency
   0–4	0.2
   5–9	0.35
   10–14	0.25
   15–19	0.20

   (a) Check the table totals 1. (b) If there were 200 buses observed, how many arrived in the 5–9 minute interval? (c) What is the probability that a bus arrives in under 10 minutes?

Challenge Questions

These problems combine grouped frequency tables with estimation, interpolation, and probability-style reasoning. They are designed to reflect the hardest GCSE exam questions.

1. The table shows the speeds of 80 cars on a motorway:

   Speed (km/h)	Frequency
   60–69	6
   70–79	14
   80–89	28
   90–99	22
   100–109	10

   (a) Estimate the mean speed. (b) Find the median speed using interpolation. (c) Identify the modal class and explain why.
2. A survey of daily screen time (hours) for 100 teenagers is summarised:

   Hours	Frequency
   0–1	8
   2–3	22
   4–5	35
   6–7	25
   8–9	10

   (a) Draw a cumulative frequency table. (b) Estimate the median and interquartile range (IQR). (c) Use the data to comment on how typical a 5-hour screen time is.
3. The waiting times (minutes) at a hospital are recorded for 120 patients:

   Time (mins)	Frequency
   0–14	15
   15–29	25
   30–44	30
   45–59	28
   60–74	22

   (a) Estimate the mean waiting time. (b) Use interpolation to find the 75th percentile. (c) What proportion of patients waited at least 45 minutes?
4. A grouped frequency table of exam marks for 200 students is given:

   Marks	Frequency
   0–19	15
   20–39	35
   40–59	80
   60–79	50
   80–99	20

   (a) Estimate the mean mark. (b) Find the median mark using cumulative frequencies. (c) Calculate the probability that a randomly chosen student scored 60 or above.
5. The lifetimes (hours) of 150 light bulbs are grouped:

   Lifetime (hours)	Frequency
   0–499	12
   500–999	30
   1000–1499	48
   1500–1999	40
   2000–2499	20

   (a) Estimate the mean lifetime. (b) Estimate the interquartile range using interpolation. (c) A bulb company advertises “average lifetime of 1600 hours”. Comment on whether the data supports this claim.

Quick Revision Sheet

Here’s a one-page summary of frequency tables for quick exam revision.

Key Definitions

• Frequency: The number of times a value occurs.
• Class interval: A range used to group continuous data (e.g. 150–159 cm).
• Midpoint: The average of the lower and upper boundaries of an interval.
• Cumulative frequency: Running total of frequencies up to each class.
• Relative frequency: Frequency ÷ total, often expressed as a probability or percentage.

Key Formulas

• Estimated mean (grouped data): \[ \bar{x} = \frac{\sum (f \times x)}{\sum f} \] where \(x\) = midpoint of class, \(f\) = frequency.
• Median (interpolation): \[ \text{Median} = L + \left(\frac{\tfrac{N}{2} - CF}{f}\right) \times w \] where: \(L\) = lower boundary of median class, \(N\) = total frequency, \(CF\) = cumulative frequency before median class, \(f\) = frequency of median class, \(w\) = class width.
• Frequency density (for histograms): \[ \text{FD} = \frac{\text{Frequency}}{\text{Class width}} \]

Steps to Build a Frequency Table

1. List all values (discrete) or intervals (grouped).
2. Tally each occurrence of the data.
3. Write the total frequency for each row.
4. Add a cumulative frequency column if needed.
5. Check that the total frequency matches the number of data points.

Common Traps

• Overlapping intervals (e.g. 10–20, 20–30). Use 10–19, 20–29 instead.
• Forgetting midpoints when estimating the mean.
• Confusing modal value (discrete) with modal class (grouped).
• Not checking that total frequencies add up correctly.

Quick Practice

1. A grouped frequency table of test scores has total frequency 50. The cumulative frequency before the median class is 20, the class frequency is 15, the lower boundary is 40, and the class width is 10. Estimate the median. Answer: \(40 + \big(\tfrac{25 - 20}{15}\big)\times 10 = 43.3\).
2. A class interval 150–159 has frequency 12. Find the frequency density. Answer: \(12 ÷ 10 = 1.2\).

Conclusion & Next Steps

Frequency tables are a cornerstone of GCSE Statistics. They organise raw data into a structured format, making it easier to calculate averages, identify trends, and link to graphs such as bar charts, histograms, and cumulative frequency curves.

• What you’ve mastered: constructing discrete and grouped tables, using midpoints for estimated means, applying cumulative frequencies for medians, and identifying modal values or classes.
• Exam habits: always total your frequencies, double-check class intervals don’t overlap, and show all steps when estimating averages (method marks matter).
• Sanity checks: totals must equal the sample size, means must fall within the data range, and frequencies should always be whole numbers.

Where this connects next:

1. Histograms: Use frequency tables with class widths to construct frequency density diagrams.
2. Cumulative Frequency Graphs: Extend tables with cumulative totals to estimate medians and quartiles visually.
3. Box Plots: Combine quartiles from cumulative frequency tables into a five-number summary diagram.
4. Probability: Interpret frequencies as probabilities by dividing by the total — useful in probability and sampling questions.
5. Comparisons: Use frequency tables to compare two sets of data side by side, commenting on averages and spreads.