Introduction: Changing Your Data

In Statistics, data sets are rarely static. Sometimes you realize a mistake was made and a value needs to be removed, or a new person joins a survey and their data must be added. Other times, you might want to change every piece of data at once—for example, giving everyone in a class a 5-mark bonus on a test or converting prices from pounds to pence.

In this chapter, we explore how these changes affect the measures of central tendency: the mean, median, and mode. Understanding these rules saves you from having to re-calculate everything from scratch!

1. Adding or Removing Data

When you add or remove a single value, the "balance" of the data set changes. The arithmetic mean is usually the most affected by these changes.

Adding a Value

Think of the mean as a see-saw. Where you add weight determines which way it tips:

  • If the new value is greater than the current mean, the mean will increase.
  • If the new value is smaller than the current mean, the mean will decrease.
  • If the new value is exactly equal to the mean, the mean stays the same.

Removing a Value

This works in the opposite way:

  • Removing a value higher than the mean will pull the mean down.
  • Removing a value lower than the mean will push the mean up.

Quick Example:
Suppose a group of 5 friends has a mean height of \(160 \text{ cm}\).
If a new friend joins who is \(180 \text{ cm}\) tall (taller than the mean), the new mean for the 6 friends must be higher than \(160 \text{ cm}\).

Key Takeaway: Adding or removing values near the "middle" has a small effect, but adding or removing outliers (extreme values) has a much larger effect on the mean.

2. Transforming Data: Translations

A translation is a fancy word for adding or subtracting the same constant value to every single item in a data set.

The Rule: If you add \(k\) to every value, the mean, median, and mode all increase by \(k\). If you subtract \(k\), they all decrease by \(k\).

Analogy: Imagine a class of students standing in a line based on their height. If every student stands on a step that is \(10 \text{ cm}\) high, their "average" height increases by exactly \(10 \text{ cm}\). Their relative positions don't change, so the median and mode move by the same amount too!

Example:

A set of test scores has a mean of \(45\), a median of \(44\), and a mode of \(42\). The teacher decides to add a "bonus" of \(5\) marks to every student's score.

  • New Mean: \(45 + 5 = 50\)
  • New Median: \(44 + 5 = 49\)
  • New Mode: \(42 + 5 = 47\)

3. Transforming Data: Scaling

Scaling involves multiplying or dividing every value in a data set by the same constant factor. This often happens during unit conversions (e.g., changing meters to centimeters).

The Rule: If you multiply every value by a factor of \(a\), the mean, median, and mode are all multiplied by \(a\).

Example:

A shop records the prices of items in pounds (\(£\)). The mean price is \(£1.20\). The shopkeeper wants to convert all prices to pence (\(p\)). Since there are \(100\) pence in a pound, we multiply every value by \(100\).

New Mean: \(1.20 \times 100 = 120\text{p}\)

Common Mistake to Avoid: Students sometimes forget that all averages (mean, median, and mode) are affected by scaling and translation. Don't just change the mean!

4. Combined Transformations (Higher Tier)

Sometimes, you might perform both a translation and scaling at the same time. This follows the basic rules of algebra: apply the multiplication/division first, then the addition/subtraction (or vice versa, depending on the context of the question).

The Formula: If the original data is \(x\) and the transformed data is \(y = ax + b\):
\(\text{New Average} = (a \times \text{Old Average}) + b\)

Step-by-Step Process:
  1. Identify the old measure of central tendency (mean, median, or mode).
  2. Identify what is being multiplied or divided (the scaling).
  3. Identify what is being added or subtracted (the translation).
  4. Apply both to the old average to find the new one.

Example:
The mean temperature in a city is \(20^\circ\text{C}\). To convert this to degrees Fahrenheit (\(^\circ\text{F}\)), the formula is \(F = 1.8C + 32\).
New Mean: \((1.8 \times 20) + 32 = 36 + 32 = 68^\circ\text{F}\).

Summary Checklist

  • Adding data > Mean: Mean increases.
  • Adding data < Mean: Mean decreases.
  • Adding \(k\) to all data: Mean, Median, and Mode all increase by \(k\).
  • Multiplying all data by \(k\): Mean, Median, and Mode are all multiplied by \(k\).

Quick Review: If you double every number in a list and then add 1, what happens to the median?
Answer: The median will be doubled and then increased by 1!