Welcome to Expected Frequency and Risk!
Have you ever wondered how medical researchers determine whether a new medicine works, how insurance companies calculate premiums, or how casinos know how much money they will make over thousands of roulette spins? The secret lies in two closely related statistical ideas: Expected Frequency and Risk.
Don't worry if this seems a bit daunting at first! We will break down every idea step-by-step with clear examples so you can tackle any question in your CCEA GCSE Statistics exam with total confidence.
---1. Expected Frequency
What is Expected Frequency?
The expected frequency is the theoretical number of times you predict an event will happen over a set number of independent trials or observations.
Think of it like this: if you flip a fair coin \(10\) times, you know the probability of getting a Head is \(\frac{1}{2}\). How many Heads do you expect to get? You naturally multiply \(10 \times \frac{1}{2} = 5\). That is all expected frequency is!
The Formula
To calculate expected frequency, use the simple formula:
\(\text{Expected Frequency} = n \times P(A)\)
Where:
• \(n\) = the total number of trials, sample size, or population size.
• \(P(A)\) = the probability (theoretical probability or experimental relative frequency) of event \(A\) occurring.
Step-by-Step Worked Examples
Example 1: Whole Number Answer
A fair six-sided die is rolled \(120\) times. How many times would you expect to roll a \(4\)?
• Step 1: Find the probability of the event. For a fair die, \(P(\text{rolling a } 4) = \frac{1}{6}\).
• Step 2: Identify the number of trials: \(n = 120\).
• Step 3: Multiply them: \(\text{Expected Frequency} = 120 \times \frac{1}{6} = 20\).
You would expect to roll a \(4\) exactly \(20\) times.
Example 2: Decimal Answer (Crucial Exam Concept!)
The probability that a bus arrives on time at a particular stop is \(0.35\). If you monitor the bus for \(50\) days, what is the expected number of days it arrives on time?
• Step 1: \(P(\text{on time}) = 0.35\).
• Step 2: \(n = 50\).
• Step 3: \(\text{Expected Frequency} = 50 \times 0.35 = 17.5\).
Important Note: Keep your answer as \(17.5\)! Do not round it to \(18\). In statistics, an expected frequency does not have to be a whole number.
Two Golden Rules of Expected Frequency
1. Expected frequencies can be decimals: If you flip a coin \(5\) times, the expected number of heads is \(5 \times 0.5 = 2.5\). Even though you cannot flip half a head in real life, \(2.5\) is the correct mathematical average expectation.
2. The Law of Large Numbers (Long-run relative frequency): If you only roll a die \(6\) times, you might not get each number once. But if you roll it \(60,000\) times, the actual observed frequency will get closer and closer to the theoretical expected frequency.
Key Takeaway: Multiply the total number of trials by the probability (\(n \times P(A)\)), and never round off a decimal answer unless specifically instructed!
---2. Understanding Risk in Statistics
In everyday language, "risk" sounds like something dangerous (like skydiving). In statistics, however, risk simply means the probability or proportion of individuals in a specific group who experience a particular outcome or event (which could be catching a cold, passing a test, or having a car breakdown).
A. Absolute Risk (AR)
Absolute Risk is the likelihood of an event occurring within one specific group.
Formula:
\(\text{Absolute Risk} = \frac{\text{Number of individuals in the group experiencing the event}}{\text{Total number of individuals in that group}}\)
• Absolute risk always lies between \(0\) and \(1\).
• You can write it as a fraction, a decimal, a percentage, or a rate such as "1 in \(N\)".
Example of Absolute Risk:
In a group of \(200\) patients who took a new allergy tablet, \(14\) experienced drowsiness.
\(\text{Absolute Risk of drowsiness} = \frac{14}{200} = 0.07 \text{ (or } 7\%\text{)}\).
B. Relative Risk (RR)
Relative Risk compares the absolute risk of an event in one group (such as a group exposed to a treatment or condition) to the absolute risk in another group (such as an unexposed or control group).
Formula:
\(\text{Relative Risk (RR)} = \frac{\text{Absolute Risk in Exposed / Group 1}}{\text{Absolute Risk in Unexposed / Group 2}}\)
How to Interpret Relative Risk Values
Exam questions will frequently ask you to explain what your calculated value of \(RR\) means. Memorise these three standard thresholds:
• \(RR = 1\): The risk is identical in both groups. There is no association between the condition/exposure and the outcome.
• \(RR > 1\): The risk is higher in Group 1 than in Group 2 (increased risk / positive association). For example, \(RR = 2.5\) means Group 1 is \(2.5\) times as likely to experience the event.
• \(RR < 1\): The risk is lower in Group 1 than in Group 2 (reduced risk / protective factor). For example, \(RR = 0.4\) means the risk in Group 1 is less than half the risk in Group 2.
Key Takeaway: Absolute Risk looks at one group alone. Relative Risk divides one group's risk by another to compare them.
---3. Working with Two-Way \((2 \times 2)\) Contingency Tables
In CCEA GCSE Statistics exams, risk questions are commonly presented inside a \(2 \times 2\) table. Let's look at how the data is structured:
Standard Table Layout:
• Group 1 (Exposed / Treatment): Event Occurs = \(a\), Event Does Not Occur = \(b\), Group Total = \(a + b\)
• Group 2 (Unexposed / Control): Event Occurs = \(c\), Event Does Not Occur = \(d\), Group Total = \(c + d\)
Using this table:
• \(\text{Absolute Risk (Exposed)} = \frac{a}{a + b}\)
• \(\text{Absolute Risk (Unexposed)} = \frac{c}{c + d}\)
• \(\text{Relative Risk} = \frac{\frac{a}{a + b}}{\frac{c}{c + d}}\)
Full Step-by-Step Exam-Style Example
A medical trial investigated whether taking a daily vitamin supplement reduces the risk of catching the winter flu. The results are shown below:
• Vitamin Group (Exposed): Flu = \(12\), No Flu = \(88\), Total = \(100\)
• Placebo Group (Unexposed): Flu = \(30\), No Flu = \(70\), Total = \(100\)
Question (a): Find the absolute risk of catching flu for those taking vitamins.
Answer: \(\text{Absolute Risk (Vitamin)} = \frac{12}{100} = 0.12\)
Question (b): Find the absolute risk of catching flu for the placebo group.
Answer: \(\text{Absolute Risk (Placebo)} = \frac{30}{100} = 0.30\)
Question (c): Calculate the relative risk of catching flu for the vitamin group compared to the placebo group.
Answer: \(\text{Relative Risk} = \frac{0.12}{0.30} = 0.4\)
Question (d): Interpret your answer to part (c).
Answer: Since \(RR = 0.4 < 1\), the risk of catching the flu is lower for those taking the vitamin supplement compared to the placebo group. The vitamin acts as a protective factor.
4. Common Exam Pitfalls & Examiner Warnings
Make sure you don't lose easy marks to these frequent mistakes highlighted by examiners:
1. Denominator Errors in Tables:
Always divide by the row total (group total), not the column total or the grand total! When calculating the risk for the treatment group, the denominator must be \(a + b\), never \(a + c\) or the overall total.
2. Rounding Expected Frequencies:
Do not round an expected frequency of \(12.5\) to \(13\). Write down the exact decimal unless the question asks for rounding.
3. Confusing Absolute Risk with Relative Risk:
Remember: Absolute risk is a single probability (\(\frac{\text{event}}{\text{total}}\)). Relative risk is a ratio of two risks (\(\frac{\text{Risk 1}}{\text{Risk 2}}\)). Never give a difference (like \(0.30 - 0.12 = 0.18\)) when asked for Relative Risk.
4. Flipping the Relative Risk Ratio:
Be careful which group goes on top! Unless told otherwise, the test/exposed group is in the numerator, and the control/unexposed group is in the denominator.
5. Quick Summary & Revision Checklist
Check off these essential skills before your exam:
• Can you calculate expected frequency using \(\text{Expected Frequency} = n \times P(A)\)?
• Do you remember that expected frequencies can be written as decimals?
• Can you calculate Absolute Risk from a group total?
• Can you calculate Relative Risk by dividing two absolute risks?
• Can you interpret whether \(RR = 1\), \(RR > 1\), or \(RR < 1\)?