A frequency table turns a list of results into four columns that answer different questions. The relative frequency column is also your best estimate of the underlying probability.
Today's hook, When you cannot calculate a probability from theory, you run the experiment and count. Relative frequency is how the counting becomes an estimate, and more trials make it better.
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You roll a die 60 times and get a six 13 times. What is your best estimate of the probability of a six? Is it $\frac{1}{6}$, or $\frac{13}{60}$, or something else?
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
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Four columns, four questions
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Four columns, four questions
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A frequency table lists each value with how often it occurred (frequency), what share of the total that is (relative frequency), the running total (cumulative frequency) and the running share (cumulative relative frequency). The relative frequency column doubles as an estimate of probability.
relative frequency $= \dfrac{\text{frequency}}{\text{total}}$ cumulative: running total down the column last cumulative relative frequency $= 1$
Relative frequencies sum to 1
Every result falls into exactly one row, so the shares must total 1. It is the fastest check that the table is right.
Cumulative means running total
Each cumulative entry is the one above it plus the current frequency. The last one equals the number of trials.
More trials, better estimate
Relative frequency approaches the true probability as the number of trials grows. A small sample can be a long way off.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
Frequency counts how many times each value occurred.
Relative frequency is frequency divided by the total number of results.
Cumulative frequency is the running total, ending at the number of trials.
Relative frequency estimates the probability of a result in an experiment.
Understand
Concepts
Why relative frequencies must sum to 1.
Why the final cumulative frequency equals the number of trials.
Why relative frequency is only an estimate, and why more trials improve it.
Can do
Skills
Build a complete four-column frequency table from raw data.
Use relative frequency to estimate a probability.
Explain why an experimental estimate differs from a theoretical probability.
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Key terms
FrequencyHow many times a particular value occurred. Like this: if a six came up 13 times in 60 rolls, its frequency is 13.
Relative frequencyThe frequency divided by the total number of results, giving a share of the whole. Like this: $\frac{13}{60} \approx 0.217$.
Cumulative frequencyA running total of the frequencies down the table. Like this: frequencies 5, 8, 7 give cumulative values 5, 13, 20.
Cumulative relative frequencyA running total of the relative frequencies, ending at 1. Like this: 0.25, 0.65, 1.00.
Experimental probabilityA probability estimated by running trials and counting, rather than by theory. Like this: 13 sixes in 60 rolls estimates the probability as about 0.217.
Theoretical probabilityThe probability worked out from the structure of the experiment. Like this: a fair die gives $\frac{1}{6} \approx 0.167$ for a six.
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Building the table
Work through the core explanation before applying it.
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Building the table
core concept
List each distinct value in the first column and count how often it occurred to get the **frequency**. The frequencies must add to the number of results.
**Relative frequency** is each frequency divided by the total. Because every result is counted in exactly one row, these must sum to 1, which is the quickest check on the whole table.
**Cumulative frequency** is the running total: each entry is the previous cumulative value plus the current frequency. The final entry equals the total number of results. **Cumulative relative frequency** is the same idea on the shares, so it ends at 1.
Rounding can break the check. If you round relative frequencies to two decimal places they may total 0.99 or 1.01. Keep full accuracy in the working and round only when presenting.
Quick check: in 40 trials a value occurred 10 times. What is its relative frequency?
Frequency counts occurrences, relative frequency divides by the total, cumulative frequency is the running total ending at the number of trials, and cumulative relative frequency ends at 1. Relative frequencies summing to 1 is the check that the table is correct.
Pause, copy the four column definitions, a small worked table, and the sum-to-1 check with the rounding warning, into your book.
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Relative frequency as a probability estimate
core concept
We just saw how to organise results. That raises a question: what can the table actually tell you about the experiment, rather than just about the trials you happened to run? This card answers it → the relative frequency column is your best estimate of the underlying probability.
When a probability cannot be calculated from theory, you run trials and count. The relative frequency of a result is the estimate: 13 sixes in 60 rolls estimates $P(\text{six}) \approx \frac{13}{60} \approx 0.217$.
It is an **estimate**, not the true value. A fair die has $P(\text{six}) = \frac{1}{6} \approx 0.167$, and 0.217 differs from that simply because 60 rolls is a small sample.
The more trials you run, the closer the relative frequency tends to sit to the true probability. That is why 6000 rolls gives a far more trustworthy estimate than 60.
A difference is not evidence of an unfair die. Getting 13 sixes in 60 rolls is entirely ordinary for a fair die. Concluding the die is biased from a small sample is a genuine error of reasoning, not just imprecision.
Fill the blank: in 200 trials an outcome occurred 50 times, so its relative frequency is 0..
Relative frequency estimates the probability of a result when theory cannot supply it. It is an estimate, so it will differ from any theoretical value, and the difference shrinks as the number of trials grows. A small-sample difference is not evidence of bias.
Pause, copy the estimate from the dice example, the comparison with the theoretical $\frac{1}{6}$, and the warning against concluding bias from a small sample, into your book.
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Reading the cumulative columns
core concept
We just saw what a single row tells you. That raises a question: what are the two cumulative columns actually for? This card answers it → they answer "how many at most" questions in one reading.
Cumulative frequency answers "how many results were this value **or less**". If the cumulative frequency beside the value 3 is 20, then 20 results were 3 or lower.
Cumulative relative frequency does the same as a proportion. A cumulative relative frequency of 0.65 beside the value 3 means 65 per cent of results were 3 or lower.
The final entries are fixed: cumulative frequency ends at the number of trials, and cumulative relative frequency ends at 1. If either does not, the table has an arithmetic error.
Cumulative columns depend on the order of the rows. They only mean "this value or less" when the values are listed in increasing order. Shuffle the rows and the cumulative columns become meaningless.
Which statement about a completed frequency table is FALSE?
Cumulative frequency answers "this value or less" as a count, cumulative relative frequency as a proportion. The final entries must equal the number of trials and 1 respectively, which checks the arithmetic. The columns are only meaningful when the values are in increasing order.
Pause, copy what each cumulative column answers, the two fixed final values, and the note that the rows must be in increasing order, into your book.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · BUILDING A TABLE
A spinner with values 1, 2, 3 is spun 40 times, giving 1 ten times, 2 eighteen times and 3 twelve times. Build the four-column frequency table.
A die is rolled 60 times and a six appears 13 times. Estimate the probability of a six, and compare with the theoretical value.
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Relative frequency $= \dfrac{13}{60} \approx 0.217$
The experimental estimate.
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For a fair die the theoretical value is $\dfrac{1}{6} \approx 0.167$
From the structure of the experiment.
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The estimate is higher, which is ordinary variation in $60$ trials and is not evidence of bias
More trials would tighten the estimate.
PROBLEM 3 · READING A CUMULATIVE COLUMN
In the spinner table above, what proportion of spins gave 2 or less, and how many spins was that?
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Find the row for the value $2$
Cumulative columns read down to the value in question.
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Cumulative relative frequency is $0.70$
So 70 per cent of spins gave 2 or less.
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Cumulative frequency is $28$ spins
$10 + 18 = 28$, consistent with $0.70 \times 40$.
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Quick-fire practice
Work through the core explanation before applying it.
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Quick-fire practice
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In 50 trials a result occurred 20 times. Find its relative frequency.
Frequencies are 4, 9, 7. Write the cumulative frequency column.
What must the final cumulative relative frequency equal?
A coin lands heads 46 times in 80 tosses. Estimate $P(\text{head})$.
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Revisit your dice estimate
Run the quick drill and copy the summary into your book.
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Revisit your dice estimate
At the start you were asked whether the estimate should be $\frac{1}{6}$ or $\frac{13}{60}$. Say which is the estimate from the data and which is the theoretical value, and explain what would make them agree more closely.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
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Short answer
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Short answer
ApplyBand 44 marks
Q1. A four-sided spinner is spun 50 times, giving 1 twelve times, 2 fifteen times, 3 eight times and 4 fifteen times. Construct a table showing frequency, relative frequency and cumulative frequency, and state the proportion of spins that gave 2 or less. (4 marks)
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ApplyBand 43 marks
Q2. In 200 trials of an experiment, an outcome occurred 46 times. Estimate its probability, and explain how the estimate would change if the experiment were repeated 2000 times. (3 marks)
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UnderstandBand 43 marks
Q3. A student rolls a die 30 times, gets seven sixes, and concludes the die is biased. Explain why this conclusion is not justified. (3 marks)
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📖 Comprehensive answers (click to reveal)
Practice 1: $\frac{20}{50} = 0.4$. Practice 2: 4, 13, 20. Practice 3: 1. Practice 4: $\frac{46}{80} = 0.575$.
Q1 (4 marks): Frequencies 12, 15, 8, 15 totalling 50 [1]. Relative frequencies 0.24, 0.30, 0.16, 0.30, summing to 1 [1]. Cumulative frequencies 12, 27, 35, 50 [1]. Spins giving 2 or less: cumulative frequency 27, that is $\frac{27}{50} = 0.54$, so 54 per cent [1].
Q2 (3 marks): Relative frequency $= \frac{46}{200} = 0.23$, so the estimated probability is 0.23 [1]. With 2000 trials the relative frequency would tend to sit closer to the true probability [1]. The estimate would therefore be more reliable, though not necessarily equal to 0.23, since a larger sample reduces the expected size of the difference [1].
Q3 (3 marks): Seven sixes in 30 rolls is a relative frequency of about 0.233, above the theoretical $\frac{1}{6} \approx 0.167$ [1]. However, 30 trials is a small sample, and this much variation is entirely ordinary for a fair die [1]. Relative frequency only approaches the true probability as the number of trials grows, so a conclusion of bias would need far more rolls before it could be supported [1].
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Boss battle · Frequency Fixer
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Build frequency tables, compute relative and cumulative columns, and estimate probabilities. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.