Linear Transformations of Data

Introduction: Why changing data matters 📊

students, imagine a class test where every student’s score is increased by $5$ marks because the teacher adds a bonus question. Or imagine a coach who multiplies every runner’s time by $0.9$ to model a faster track. These are examples of linear transformations of data. In statistics, a linear transformation changes every data value using a rule of the form $y=ax+b$, where $a$ and $b$ are constants.

This lesson helps you understand how data changes when we add, subtract, multiply, or divide by the same number. You will learn how these transformations affect measures such as the mean, median, quartiles, range, and standard deviation. You will also see why this matters in real life, from exam scores to temperature conversion and financial scaling. 🌍

Learning objectives

Explain the main ideas and terminology behind linear transformations of data.
Apply IB Mathematics: Analysis and Approaches HL reasoning and procedures related to linear transformations of data.
Connect linear transformations of data to statistics and probability.
Summarize how linear transformations of data fit within statistics and probability.
Use examples to support understanding of linear transformations of data.

A key idea is this: some statistics change predictably when data is transformed, and some do not. Understanding this lets you interpret data correctly instead of being misled by a new scale or unit.

What is a linear transformation of data?

A linear transformation applies the same algebraic rule to every value in a data set. If the original value is $x$, the transformed value is $y=ax+b$.

There are two common parts:

Multiplication by a constant: $y=ax$
Addition of a constant: $y=x+b$

Together they give the full form $y=ax+b$.

For example, suppose the original data are test scores:

$40, 50, 60, 70

If every score is increased by $5$, the transformed data are:

$45, 55, 65, 75

If every score is doubled, the transformed data are:

$80, 100, 120, 140

If every score is transformed by $y=2x+5$, then the new data are:

$85, 105, 125, 145

This is not just changing numbers randomly. The relationship between old and new values is consistent for every item in the set.

Important terminology

Original data: the data before transformation.
Transformed data: the data after applying the rule.
Scale factor: the multiplier $a$ in $y=ax+b$.
Shift or translation: the constant $b$ in $y=ax+b$.
Positive transformation: when $a>0$, the order of values stays the same.
Negative transformation: when $a<0$, the order reverses.

In IB Mathematics: Analysis and Approaches HL, you should be able to describe how a transformation affects statistical summaries and interpret data in context.

What happens to measures of centre and spread?

A major part of this topic is understanding how common statistics change after a transformation. Let the original data have mean $\bar{x}$, median $m$, and standard deviation $s$.

Adding or subtracting a constant

If each data value changes from $x$ to $x+b$, then:

The mean becomes $\bar{x}+b$
The median becomes $m+b$
The quartiles become $Q_1+b$ and $Q_3+b$
The range stays the same
The interquartile range stays the same
The standard deviation stays the same

Why? Because adding the same number to every value moves the entire data set without changing how spread out it is.

Example

Data: $3, 7, 10, 12

Mean:

$\bar{x}=\frac{3+7+10+12}{4}=8$

If we add $5$ to each value, the new data are $8, 12, 15, 17.

The new mean is $13$, which is $8+5$.

The spread is unchanged because the distances between values are the same.

Multiplying by a positive constant

If each data value changes from $x$ to $ax$ with $a>0$, then:

The mean becomes $a\bar{x}$
The median becomes $am$
The quartiles become $aQ_1$ and $aQ_3$
The range becomes $a$ times the original range
The interquartile range becomes $a$ times the original IQR
The standard deviation becomes $as$

This happens because every distance from the centre is also multiplied by $a$.

Example

Data: $2, 4, 6

Mean:

$\bar{x}=\frac{2+4+6}{3}=4$

If we multiply every value by $3$, the new data are $6, 12, 18.

The new mean is $12$, which is $3\times 4$.

The standard deviation also triples.

Multiplying by a negative constant

If $a<0$, then the data are reflected and scaled. The order of values reverses.

For example, if $x$ becomes $-2x$, then larger original values become smaller transformed values.

The mean becomes $a\bar{x}$
The median becomes $am$
The standard deviation becomes $|a|s$

The standard deviation is always non-negative, so we use the absolute value of the multiplier.

How does a linear transformation affect graphs and box plots? 📈

Visual representations help you see the effect of transformation.

On a scatter graph

If a data set is transformed using $y=ax+b$, then every point is moved according to the rule. The pattern often keeps the same overall shape if $a>0$, but the scale changes.

For example, if height is measured in centimetres and then converted to millimetres, each value is multiplied by $10$. The graph looks the same in shape, but the axis numbers change.

On a box plot

A box plot shows the median, quartiles, and possible outliers. Under a linear transformation:

Adding a constant shifts the whole box plot left or right, or up or down.
Multiplying by a positive constant stretches or compresses the box plot.
If the multiplier is negative, the box plot is reflected and reversed.

This is why box plots are useful for comparing transformed data. You can often predict the new plot without recalculating every value.

Real-world example

Suppose the temperatures in a week are recorded in degrees Celsius. To convert to degrees Fahrenheit, use

$F=\frac{9}{5}C+32$

This is a linear transformation. The addition of $32$ shifts the scale, and the multiplier $\frac{9}{5}$ stretches the values. The order of temperatures stays the same because the multiplier is positive.

Why does this matter in statistics and probability?

Linear transformations are important because data in the real world often come in different units or scales. Statistics must still be interpreted correctly.

Example 1: Test scores

If a teacher scales a quiz by adding $10$ bonus points to every student, the relative differences remain the same. The class average increases by $10$, but the spread does not change. This means the class became more generous in marking, not more varied in ability.

Example 2: Money and currency

If a price is converted from pounds to euros using a fixed exchange rate, the data are multiplied by a constant. Measures like mean and standard deviation scale too.

Example 3: Science measurements

In physics or chemistry, changing units often creates linear transformations. For example, converting centimetres to metres uses multiplication by $\frac{1}{100}$. The actual physical quantity is unchanged, but the numerical representation changes.

Connection to probability distributions

Although this lesson focuses on data, linear transformations also matter for random variables. If $X$ is a random variable and $Y=aX+b$, then the mean and standard deviation transform in the same way:

$E(Y)=aE(X)+b$
$\mathrm{SD}(Y)=|a|\mathrm{SD}(X)$

This connects statistics to probability, because the same rules apply to distributions as well as data sets.

Worked example: interpreting transformed data

Suppose the times in seconds for $5$ students to finish a puzzle are:

$12, 15, 18, 20, 25

Now each time is transformed using $y=1.5x+2$.

The transformed values are:

$1.5(12)+2=20$
$1.5(15)+2=24.5$
$1.5(18)+2=29$
$1.5(20)+2=32$
$1.5(25)+2=39.5$

Original mean:

$\bar{x}=\frac{12+15+18+20+25}{5}=18$

New mean:

$1.5(18)+2=29$

This matches the transformed values.

Original median: $18$

New median:

$1.5(18)+2=29$

Original range:

$25-12=13$

New range:

$39.5-20=19.5$

Since the multiplier is $1.5$, the new range is $1.5\times 13=19.5$.

This example shows a useful IB skill: you should be able to transform statistics directly without recalculating every single value.

Common mistakes to avoid ⚠️

Students often make predictable errors when working with transformed data.

Forgetting that adding a constant does not change spread.
Thinking the standard deviation changes when only a constant is added.
Using $a$ instead of $|a|$ for standard deviation after multiplication.
Forgetting that a negative multiplier reverses the order of the data.
Mixing up the effect on the mean and the median.

A good strategy is to ask: does the transformation change position, scale, or both?

Conclusion

Linear transformations of data are a key idea in statistics because they show how data change under common real-life operations. When data are transformed by $y=ax+b$, you can predict the new mean, median, quartiles, range, and standard deviation. Adding a constant shifts data, while multiplying by a constant rescales them. This knowledge helps you interpret graphs, compare data sets, and work with different units correctly.

For IB Mathematics: Analysis and Approaches HL, this topic builds the foundation for understanding data descriptions and links naturally to probability distributions and random variables. Mastering these rules makes it easier to analyze data confidently and accurately. ✅

Study Notes

A linear transformation of data has the form $y=ax+b$.
Adding $b$ shifts every value by the same amount.
Multiplying by $a>0$ scales all values and keeps their order.
Multiplying by $a<0$ scales values and reverses their order.
If $x$ changes to $x+b$, then the mean, median, and quartiles each increase by $b$.
If $x$ changes to $ax$ with $a>0$, then the mean, median, quartiles, range, IQR, and standard deviation are all multiplied by $a$.
If $x$ changes to $ax$ with $a<0$, the standard deviation becomes $|a|s$.
Adding a constant does not change the spread of the data.
Box plots and graphs can be transformed without recalculating every data point.
Unit conversions are often linear transformations, such as $F=\frac{9}{5}C+32$.
Linear transformations of data connect statistics to probability and random variables.
In IB Mathematics: Analysis and Approaches HL, you should be able to interpret and apply these rules in context.