Statistics
What is Statistics?
Statistics is the study of collecting, organising, and interpreting data. The four key measures that describe a data set are mean, median, mode, and range. The first three are called measures of central tendency (averages), and the last is a measure of spread.
Key terms:
- Data set: a collection of values or observations
- Average: a single value that represents the centre of a data set
- Outlier: a value that is much higher or lower than the rest of the data
Mean (Average)
The mean is calculated by adding all values together and dividing by the number of values.
Formula: Mean = Sum of all values รท Number of values
Example: Find the mean of 4, 7, 8, 5, 6
Sum = 4 + 7 + 8 + 5 + 6 = 30
Mean = 30 รท 5 = 6
The mean uses every value in the data set, which makes it sensitive to extreme values (outliers).
Median (Middle Value)
The median is the middle value when data is arranged in ascending order. It splits the data set in half.
Odd number of values: the middle number is the median.
Example: 2, 3, 5, 7, 9 โ Median = 5
Even number of values: find the mean of the two middle numbers.
Example: 3, 5, 7, 9, 11, 13 โ Median = (7 + 9) รท 2 = 8
Always sort the data into ascending order before finding the median.
Mode (Most Frequent)
The mode is the value that appears most often in the data set.
Example: 4, 7, 2, 7, 5, 3, 7 โ Mode = 7 (appears 3 times)
A data set can have:
- One mode (unimodal): 2, 3, 5, 5, 8 โ mode = 5
- Two modes (bimodal): 2, 2, 5, 5, 8 โ modes = 2 and 5
- No mode: if all values appear the same number of times
Range (Spread)
The range measures how spread out the data is.
Formula: Range = Highest value โ Lowest value
Example: 12, 5, 18, 3, 9 โ Range = 18 โ 3 = 15
A small range means data is clustered closely together. A large range means data is widely spread. Note: the range is not an average โ it measures spread, not centre.
Worked Example: All Four Measures
Data set: 6, 3, 9, 3, 7, 8, 5
Step 1 โ Sort: 3, 3, 5, 6, 7, 8, 9
Mean: (3 + 3 + 5 + 6 + 7 + 8 + 9) รท 7 = 41 รท 7 โ 5.86
Median: middle value of 7 numbers = 4th value = 6
Mode: 3 (appears twice)
Range: 9 โ 3 = 6
Choosing the Right Measure
| Use | When |
|---|---|
| Mean | Data has no extreme outliers; you need a precise average |
| Median | Data has very high or very low outliers that would skew the mean |
| Mode | You want the most common value (e.g. most popular shoe size) |
| Range | You want to understand how spread out the data is |
Classic example: A footballer scores in 5 matches: 0, 0, 1, 0, 12 goals
- Mean = 13 รท 5 = 2.6 โ misleading due to the outlier (12)
- Median = 0 โ better represents a typical week
- Mode = 0 โ also representative
- Range = 12 โ 0 = 12 โ shows the wide variation
Interquartile Range (IQR)
The interquartile range measures the spread of the middle 50% of data, making it less affected by outliers than the range.
IQR = Upper Quartile (Q3) โ Lower Quartile (Q1)
For the sorted data: 3, 5, 7, 9, 11, 13, 15
- Q1 (lower quartile) = 5
- Q3 (upper quartile) = 13
- IQR = 13 โ 5 = 8
Common Mistakes
- Forgetting to sort data before finding the median
- Confusing the median position for an even set (average the two middle values, do not just pick one)
- Stating there is no mode when every value is different (correct) vs when values repeat equally (bimodal)
- Using the mean when an outlier makes it unrepresentative
- Calculating the range as highest + lowest instead of highest โ lowest
Tips and Tricks
- Mean: "add them all up and divide" โ the most commonly required average
- Median: think "middle" โ always sort first
- Mode: think "most" โ the most frequently occurring value
- For a large sorted list, the median is at position (n + 1) รท 2
- When comparing two data sets, compare both an average (mean or median) and a spread measure (range or IQR)