Descriptive Analysis and Descriptive Statistics
Your class just sat a test and there are 30 scores. Reading all 30 tells you almost nothing at a glance. But three small numbers, the average, the middle and the spread, tell the whole story instantly.
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Two friends each grew five tomato plants and counted the tomatoes on each plant. Friend A got 4, 7, 7, 9 and 13. Friend B got 8, 8, 8, 8 and 8.
If you had to describe each friend's results using just one or two numbers, which numbers would you choose, and why?
Descriptive statistics are numbers that summarise a dataset. They take a long list of values and squeeze it down to a few figures that describe what the data looks like. This is what we mean by descriptive analysis: describing the data you actually collected, rather than guessing about data you do not have.
It is important not to confuse two different jobs. Descriptive statistics only describe the data in front of you: the typical value, the spread, the most common result. They do not predict the future or infer what would happen to people you never measured. Saying "the average mark in this class was 72" is descriptive. Saying "the average mark for all Year 9 students in Australia is 72" would be an inference, and needs different, more careful methods. In this lesson we stay with the describing job.
A scientist measures the daily rainfall in Sydney for a whole year, giving 365 numbers. Nobody wants to read all 365. Instead they report descriptive statistics: the average daily rainfall, the wettest and driest day, and the most common amount. Three or four numbers describe the entire year.
The Australian Bureau of Statistics (ABS) collects data from millions of people, then publishes a handful of descriptive statistics, such as the typical income or the typical house price, so the whole country can understand the result in seconds.
Descriptive statistics never tell the whole story. They are a useful summary, but two very different datasets can share the same average. Always look at the spread as well as the centre.
Know
- What descriptive statistics are and how they differ from prediction or inference.
- The meaning of mean, median, mode and range.
Understand
- Why the median can be more reliable than the mean when there are outliers.
- How a frequency table organises data before you summarise it.
Can Do
- Calculate the mean, median, mode and range of a dataset.
- Choose the best summary number for a given set of data.
The most useful summary is usually the centre of the data, the typical or middle value. There are three different ways to measure the centre, and they are called measures of central tendency.
The mean is the sum of all the values divided by the number of values. The median is the middle value once you put the data in order. The mode is the value that appears most often. Let us work through one dataset carefully: 4, 7, 7, 9, 13.
For the mean, add the values: 4 + 7 + 7 + 9 + 13 = 40. There are 5 values, so the mean is 40 ÷ 5 = 8. For the median, the data is already in order, so the middle value (the 3rd of 5) is 7. For the mode, the value 7 appears twice and every other value appears once, so the mode is 7.
If a dataset has an even number of values, there is no single middle value, so the median is the average of the two middle values. For 3, 5, 8, 10 the two middle values are 5 and 8, so the median is (5 + 8) ÷ 2 = 6.5.
The number of pets owned by 7 students is 0, 1, 1, 1, 2, 3, 5. Mean = (0 + 1 + 1 + 1 + 2 + 3 + 5) ÷ 7 = 13 ÷ 7 = 1.9 (to 1 decimal place). Median = the 4th value = 1. Mode = 1 (it appears three times). Range = 5 − 0 = 5.
Before you find a median, you must put the values in order. Picking the middle of an unsorted list is a very common mistake and gives the wrong answer.
Knowing the centre is only half the story. Two datasets can have the same mean but look completely different. The spread tells you how varied or scattered the data is. The simplest measure of spread is the range: the highest value minus the lowest value.
Think back to the tomato plants. Friend A got 4, 7, 7, 9, 13 and Friend B got 8, 8, 8, 8, 8. Both have a mean of 8, but Friend A's range is 13 − 4 = 9, while Friend B's range is 8 − 8 = 0. The means are identical, yet the spreads could not be more different. Friend B's plants were perfectly consistent; Friend A's varied a lot. The range captures that difference in a single number.
Weather scientists at the Bureau of Meteorology report both the average daily temperature and the temperature range. A day with a mean of 22 degrees but a range from 8 to 36 degrees feels nothing like a steady 22 all day, the spread matters.
Do not confuse the range with the general idea of "spread". The range is one specific number (highest minus lowest). It is sensitive to outliers, because a single extreme value can make the range huge.
A student works out the statistics for the dataset 5, 9, 9, 12. One line is wrong, click it.
- Mean = (5 + 9 + 9 + 12) ÷ 4 = 35 ÷ 4 = 8.75
- Range = 12
- Mode = 9 (it appears twice, more than any other value)
The mean is the most familiar average, but it has a weakness: it is easily dragged around by an outlier, a value much larger or smaller than the rest. The median is far more stable, because it only cares about the middle position, not the actual size of the extreme values.
Watch what happens to our dataset 4, 7, 7, 9, 13 (mean 8, median 7) when we add one outlier of 60. The new dataset is 4, 7, 7, 9, 13, 60. The new mean is (4 + 7 + 7 + 9 + 13 + 60) ÷ 6 = 100 ÷ 6 = 16.7 (to 1 decimal place). The mean has leapt from 8 to 16.7, even though only one value changed. The new median is the average of the two middle values (the 3rd and 4th of 6), which are 7 and 9, so the median is (7 + 9) ÷ 2 = 8. The median barely moved, from 7 to 8.
This is exactly why, when data is skewed or contains outliers, the median often describes the typical value far better than the mean. The mean now suggests a "typical" value of 16.7, but five of the six values are 13 or below, so the median of 8 is a much fairer summary.
This is why the ABS reports the median house price and the median income, not the mean. A few billionaires or a handful of multimillion-dollar mansions would pull the mean far above what a typical Australian earns or pays. The median ignores those extremes and reports the genuine middle, a fairer picture of normal life.
The median is not always the "right" answer. For symmetric data with no outliers, the mean is perfectly good and uses every value. Choose the median when outliers or skew would make the mean misleading.
Before you summarise messy data, it helps to organise it. A frequency table lists each value alongside its frequency, how many times that value appears. Suppose 9 students score 6, 7, 7, 8, 8, 8, 9, 9, 10 out of 10. The frequency table groups them: a score of 6 appears once, 7 appears twice, 8 appears three times, 9 appears twice, and 10 appears once. From the table the mode is instantly clear (8, the highest frequency), and the mean is easy to compute: (6×1) + (7×2) + (8×3) + (9×2) + (10×1) = 6 + 14 + 24 + 18 + 10 = 72, divided by 9 students = 8.
So why bother with descriptive statistics at all? The benefits are powerful. They condense a huge dataset into a few numbers anyone can grasp. They allow fair comparison between groups, you can compare two classes by their medians even if the classes are different sizes. They reveal both the typical value and the spread at a glance. And they let scientists communicate findings quickly, in a headline, a graph caption, or a single sentence, instead of forcing everyone to read raw data.
Two Year 9 classes sit the same quiz. Class P has 18 students and Class Q has 25. You cannot compare them by adding up totals, because the classes are different sizes. But the median score of each class can be compared directly, that is the power of a good summary statistic.
When you find the mean from a frequency table, remember to multiply each value by its frequency before adding, and divide by the total frequency (9 here), not by the number of different scores (5 here).
A small town has 9 households. Eight earn about $60,000 a year, but one household earns $5,000,000. A reporter wants to describe the "typical" income. Should they use the mean or the median, and roughly what will each value be?
How close was your prediction?
Nice, you spotted that the outlier wrecks the mean and the median is the fair choice.
Good to notice, one extreme value pulls the mean far away from the typical household.
Speed Round · 6 questions
True or false? Tap as fast as you can. Build a streak.
The mean is found by adding all values and dividing by how many values there are.
The median is always the same as the mean.
The range is the highest value minus the lowest value.
An outlier affects the median more than it affects the mean.
The mode is the value that appears most often.
Descriptive statistics predict what will happen to data you have not collected.
How are you completing this lesson?
Think back to the two friends' tomato plants: 4, 7, 7, 9, 13 and 8, 8, 8, 8, 8.
Both have a mean of 8. Which one number would you now add to describe the difference between them, and what does it tell you?
Quick Check · 5 questions
Check Your Understanding · 3 questions
1. In your own words, explain what descriptive statistics are and give one example of a question they answer about a dataset.
2. For the dataset 2, 4, 4, 6, 9, calculate the mean, the median, the mode and the range. Show your working.
3. Two classes get the same mean score on a test. Explain why you would still want to know the range of each class before deciding they performed the same.
Show Your Working · 3 questions
SA1. A student records the number of goals scored in 6 netball games: 3, 5, 5, 8, 10, 11. Calculate the mean, median, mode and range, showing your working for each.
SA2. A dataset is 6, 7, 7, 8, 40. Calculate the mean and the median. Then explain which one better describes the typical value, and why.
Hint: Look for an outlier and think about what it does to each measure.
SA3. The ABS reports the median house price for a suburb rather than the mean. Explain, using the idea of outliers, why the median gives a fairer picture of a typical house price. Include a short numerical example to support your answer.
Quick Check
1. C. Mean = (3 + 6 + 6 + 9) ÷ 4 = 24 ÷ 4 = 6.
2. A. Ordered the data is 2, 5, 7, 8, 9; the middle (3rd) value is 7.
3. D. Range = highest − lowest = 14 − 3 = 11.
4. B. The median depends only on the middle position, so a single extreme value barely shifts it, while it can drag the mean far away.
5. C. Descriptive statistics condense a large dataset into a few numbers that allow fair comparison between groups.
Show Your Working Model Answers
SA1 (4 marks): Mean = (3 + 5 + 5 + 8 + 10 + 11) ÷ 6 = 42 ÷ 6 = 7 [1]. Median: data is in order, 6 values, so average the two middle (3rd and 4th) values = (5 + 8) ÷ 2 = 6.5 [1]. Mode = 5, because it is the only value that appears twice [1]. Range = 11 − 3 = 8 [1].
SA2 (4 marks): Mean = (6 + 7 + 7 + 8 + 40) ÷ 5 = 68 ÷ 5 = 13.6 [1]. Median: ordered the middle (3rd of 5) value is 7 [1]. The median (7) better describes the typical value [1], because 40 is an outlier that pulls the mean (13.6) far above four of the five values, which are all 8 or less [1].
SA3 (5 marks): House prices often contain a few very expensive homes that act as outliers [1]. These outliers pull the mean upwards, so the mean overstates what a typical house costs [1]. The median is the middle price once all houses are ordered, and it is barely affected by a few extreme values [1], so it better represents a typical home. Example: prices of $500,000, $550,000, $600,000, $650,000 and $5,000,000 give a mean of $7,300,000 ÷ 5 = $1,460,000 [1], but a median of $600,000, which is the fairer description of a typical house [1].
Mean
Sum of values divided by how many
Median
Middle value when data is ordered
Mode
The value that appears most often
Range
Highest value minus lowest value
Outlier
An extreme value that drags the mean
Why summarise
Condense data and compare fairly
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