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Hong KongMean, median, mode and range
The rule splits this into two skills: calculate range, mean, median and mode, and interpret central tendency and dispersion in context. The calculation is a calculator away. The interpretation is where an item can genuinely be hard, and outliers are how it gets built.
Two skills, and most preparation only covers the first. Skill 3 is the arithmetic. Skill 4 asks what the number means, and skill 5 asks whether choosing it was honest.
The four definitions
| Measure | How to find it | What it tells you |
|---|---|---|
| Mean | Add all values, divide by how many | The balance point |
| Median | Order the values, take the middle one | The typical value, ignoring extremes |
| Mode | The most frequent value | The most common case |
| Range | Largest minus smallest | How spread out the data is |
Two points on the mechanics. Order the data before you take the median, every time, and with an even number of values the median is the average of the middle two. A set can have more than one mode or none at all.
Why the mean and the median disagree
The mean uses every value, so a single extreme value drags it. The median only cares about position, so it barely moves.
Take the set 1, 2, 3, 4 and 100. The mean is 22. The median is 3. Neither is wrong and they describe different things, and an item can ask which better represents a typical value. The answer there is the median, because 22 is larger than four of the five values in the set.
A district reports the mean class size as 22 and the median as 19. What does that difference most likely indicate?
- Most classes have exactly 22 students
- A small number of unusually large classes is raising the mean
- The data must contain an error
- The range is zero
Dispersion, and what the rule includes
Skill 4 names range and standard deviation as the measures of dispersion. Range is in skill 3 as something to calculate; standard deviation is not, and it appears only in the interpretation skill.
Read that carefully, because it tells you what to expect. You are asked to interpret what a standard deviation means in context, not to compute one. A larger standard deviation means more spread. That is the level being tested.
Choosing a statistic dishonestly
Skill 5 is about exactly this: how the selection of a statistic can lead to different or inappropriate interpretations. Somebody reporting the mean salary at a company with one very high earner is choosing a number that flatters.
- Use the median when the data is skewed or has outliers
- Use the mean when the data is roughly symmetric and every value should count
- Use the mode for categories, where averaging makes no sense
A question asking which measure best represents a data set is asking you to spot the skew. Look for one value far from the rest, and if it is there, the median is almost always the answer.
What a missing value question wants
A common item gives you a mean and all but one value, and asks for the missing one. Work backwards: multiply the mean by the number of values to get the total, then subtract what you have.
It is two operations and it feels like more, which is why it is a good item. Write the total down before you subtract, because doing it in your head under a clock is where the arithmetic goes wrong even with a calculator on the screen.
Common questions
What is the difference between mean and median?
The mean adds every value and divides by the count, so extremes pull it. The median is the middle value once the data is ordered, so it barely moves. In the set 1, 2, 3, 4 and 100 the mean is 22 and the median is 3.
When should I use the median instead of the mean?
When the data is skewed or contains an outlier. The rule has a whole skill about how the selection of a statistic can mislead, and reporting a mean that sits above most of the values in the set is the classic example of that.
Do I have to calculate standard deviation on the FTCE?
The rule asks you to interpret it, not compute it. Standard deviation appears only in the skill about interpreting dispersion in context; the calculation skill names range, mean, median and mode. A larger standard deviation means more spread.
How do I find a missing value given the mean?
Multiply the mean by the number of values to get the total, then subtract the values you already have. Two operations, and writing the total down before subtracting is worth doing even with a calculator available.