Relative Frequency
Learn about relative frequency for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-07). Relative frequency is the fraction of trials in which an outcome actually happened — how often it occurred divided by the total number of trials.
You will find relative frequencies from a frequency table, use them to estimate a probability, and see that the more trials you run, the nearer they tend to get to the theoretical probability — the idea behind estimating things like the chance a product is faulty.
Theory
Relative frequency is the fraction of trials in which an outcome happened. This Year 12 Standard 1 (NSW) guide shows how to find it from a frequency table, use it as an estimate of probability, and why more trials bring it closer to the theoretical probability.
Relative frequency is the fraction of trials in which an outcome actually happened. It is found by dividing the frequency of the outcome by the total number of trials (or results). Find the total by adding every frequency in the table.
A relative frequency lies from \(0\) to \(1\) and can be written as a fraction, a decimal or a percentage. The relative frequencies of all the outcomes add to \(1\).
Relative frequency is the experimental probability: it is used to estimate a probability. As the number of trials increases, the relative frequency usually settles closer to the theoretical probability, so a large sample gives the best estimate.
Divide the frequency by the total of all the frequencies:
Use it as an estimate of the probability:
How to find a relative frequency
- Add every frequency in the table to get the total number of trials.
- Find the frequency of the outcome (add the frequencies if it is a group).
- Divide frequency by total, then give a fraction, decimal or percentage. Use it as an estimate of the probability.
| Goals | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|
| Games | \(6\) | \(12\) | \(11\) | \(7\) | \(4\) |
Add the frequencies for the total, then divide.
| \(\text{total}\) | \(=\) | \(6+12+11+7+4=40\) |
| \(\text{(i) RF}(3)\) | \(=\) | \(\dfrac{7}{40}=0.175\) |
| \(\text{(ii) fewer than }2\) | \(=\) | \(6+12=18\) |
| \(\text{RF}\) | \(=\) | \(\dfrac{18}{40}=0.45\) |
(i) \(0.175\) (ii) \(0.45\)
Use the relative frequency as the estimate.
| \(P(\text{upside-down})\) | \(\approx\) | \(\dfrac{85}{250}=0.34\) |
| \(\) | \(=\) | \(0.34\times100\%=34\%\) |
| \(P(\text{not upside-down})\) | \(\approx\) | \(1-0.34=0.66\) |
About \(0.34\) (\(34\%\)), and \(0.66\) that it does not.
| Session | Spins | Blue |
|---|---|---|
| \(1\) | \(20\) | \(8\) |
| \(2\) | \(100\) | \(29\) |
| \(3\) | \(800\) | \(196\) |
Divide blue by spins for each session.
| \(\text{Session 1: }\dfrac{8}{20}\) | \(=\) | \(0.4\) |
| \(\text{Session 2: }\dfrac{29}{100}\) | \(=\) | \(0.29\) |
| \(\text{Session 3: }\dfrac{196}{800}\) | \(=\) | \(0.245\) |
Session 3 (\(0.245\)) is closest — it has by far the most spins.
The relative frequencies add to \(1\), so bus and car together make up \(1-0.25=0.75\) of the total \(N\).
| \(18+27\) | \(=\) | \(45\) |
| \(0.75\times N\) | \(=\) | \(45\) |
| \(N\) | \(=\) | \(\dfrac{45}{0.75}=60\) |
| \(\text{walk}\) | \(=\) | \(60-45=15\) |
\(60\) students were surveyed and \(15\) walk.
Common pitfalls
Frequently asked questions
What is relative frequency?
Relative frequency is the number of times an outcome happened divided by the total number of trials. If a cap lands upside-down 85 times in 250 tosses, the relative frequency is 85 over 250, which is 0.34.
How do you calculate relative frequency from a table?
Add all the frequencies to get the total, then divide the frequency of the outcome by that total. For a group of values, add their frequencies first.
What is the difference between relative frequency and probability?
Relative frequency comes from what actually happened in an experiment. Theoretical probability is worked out from equally likely outcomes. Relative frequency is used as an estimate of probability.
Why does a larger number of trials give a better estimate?
With only a few trials the results jump around by chance. As the number of trials increases, the relative frequency usually settles closer to the theoretical probability.
Do relative frequencies add up to 1?
Yes. The relative frequencies of all the outcomes in a table add to 1, or 100 percent, because together they cover every trial.
Is relative frequency the same as experimental probability?
Yes. Relative frequency is also called experimental probability, because it is found by doing an experiment or survey rather than by reasoning about equally likely outcomes.