ACARA v9 CONTENT DESCRIPTION “acquire data sets for discrete and continuous numerical variables and calculate the range, median, mean and mode; make and justify decisions about which measures of central tendency provide useful insights into the nature of the distribution of data”
Data on its own is just a list of numbers. To make sense of it, we summarise it: we find a single value that represents the typical case, and a measure of how spread out the values are. This year you learn the four core summaries, the range, median, mean and mode, and, just as importantly, how to choose the one that tells the truest story about a data set.
These measures are everywhere, from a class average to a typical house price to the most popular shoe size. Knowing how to calculate each is the first step; knowing which one to trust in a given situation is the skill that separates careful reasoning from misleading statistics.
Centre and spread
Two questions describe almost any data set: where is its centre, and how spread out is it? The mean answers the first by sharing the total equally. Add every value and divide by how many there are, and you get the balancing point of the data. The range answers the second by subtracting the smallest value from the largest, giving a simple measure of how far the data stretches.
Centre and spread together
The mean marks the balancing centre; the range shows how widely the data spreads.
A single data set has a centre and a spread. The mean of these values is 4, the balancing point, while the range of 5 tells you how far the smallest and largest values sit apart.
The mean is powerful because it uses every value, but that is also its weakness. One unusually large or small value drags it in that direction. The range is quick to find but blunt, since it depends only on the two extreme values and ignores everything in between. Each summary captures something real and misses something else, which is why we keep several.
The median and the mode
The median is the middle value once the data is in order, so half the values sit below it and half above. The mode is simply the value that occurs most often. Both are easy to find by hand, and both resist the pull of extreme values far better than the mean, which makes them valuable when a data set has unusual outliers.
Median and mode
Order the values: the middle one is the median, the most frequent is the mode.
Once the values are in order, the median is simply the one in the middle. The mode is the value that appears most often. Here both happen to be 4, but in other data sets they can be quite different.
Choosing the right measure is the heart of this topic. For a set of house prices with one enormous mansion, the mean is pulled upward and overstates the typical price, while the median stays close to what most houses actually cost. For shoe sizes a shop should stock, the mode matters most, since it names the size that sells the most. The maths of each measure is straightforward; deciding which one represents the data honestly is the real work, and it is a judgement you will make again and again.
Teaching tip: the clearest way to show why the choice of measure matters is with a small, deliberately skewed data set. List the salaries of a few staff plus one chief executive on a huge wage, then calculate the mean and the median together. Seeing the mean land above almost every individual salary, while the median sits right among them, makes the idea of a misleading average concrete and memorable.
A frequent slip is forgetting to order the data before finding the median. Build the habit of rewriting the values in order first, every time, since the middle of an unordered list is meaningless and this is where most errors creep in.
Builds on: Comparing Data Sets (AC9M6ST01). That unit compared distributions using mode, range and shape; this unit adds the median and mean and asks which measure best represents the data.
Quick self-check
1. How do you find the mean of a set of numbers?
2. For the ordered data 3, 5, 7, 9, 11, what is the median?
3. What is the mode of the data set 4, 4, 6, 7, 4, 9?
4. What does the range of a data set measure?
5. A house price data set has one extremely expensive mansion. Which measure of centre is least distorted by it?