T.15 Simple linear regression

Construct scatter diagrams, describe correlation, find a line of best fit (y = a + bx) by eye and by least squares, use it to interpolate and extrapolate, and explain why correlation does not prove causation.

IB Business Management
Part of T Business management toolkit
HL only

Key terms introduced here

Each is in the Business glossary, with every other term in the subject.

Causation: A relationship in which a change in one variable directly brings about a change in another; correlation between two variables does not by itself show causation.
Correlation: The extent to which two variables are related, so that a change in one is associated with a change in the other; it can be positive, negative or absent, and strong or weak.
Correlation coefficient: A number between −1 and +1 (Pearson’s r) that measures the direction and strength of the linear relationship between two variables; values near ±1 show a strong relationship and values near 0 a weak one.
Dependent variable: The variable whose value is thought to depend on another variable, plotted on the vertical y-axis of a scatter diagram, such as sales revenue.
Independent variable: The variable that is thought to influence another variable, plotted on the horizontal x-axis of a scatter diagram, such as advertising spending.
Interpolation: Estimating a value within the range of the observed data, for example by reading from a line of best fit between known points.
Line of best fit

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T.15 Simple linear regression · IB Business | Augment Education

Notes on T.15 Simple linear regression

25 min readLast reviewed 15 Oct 2026

Learning objectives

Distinguish between the independent and the dependent variable in a business relationship.
Construct a scatter diagram from paired data.
Describe the direction and strength of the correlation shown by a scatter diagram or a correlation coefficient.
Draw a line of best fit by eye and calculate the least-squares line y = a + bx for a small data set.
Use a line of best fit to interpolate and extrapolate, and explain the risks of each.
Explain why correlation between two variables does not prove causation.
Apply simple linear regression to business decisions such as forecasting sales.

Introduction

Managers often want to know how one variable affects another: whether more advertising brings more sales, whether higher prices reduce the quantity sold, or whether more training raises productivity. Simple linear regression measures such a relationship from past data by fitting a straight line to it, and the line can then be used to predict one variable from the other.

Regression helps businesses respond to change by showing how sales move with incomes, prices or the economy. It also raises questions of ethics: a strong statistical link can be presented as proof that one thing causes another, and decisions about people, such as who to hire or how much to pay, should not rest on correlations that have no sound explanation.

Independent and dependent variables

Regression studies two variables at a time. One is thought to influence the other: the independent variable is the possible cause or driver, and the dependent variable is the outcome that is thought to respond to it. Deciding which is which comes from business reasoning, before any data are plotted.

In most business uses the independent variable is something the business controls or can observe in advance, such as its advertising spending, its price or the temperature forecast, and the dependent variable is something it wants to predict, such as sales, profit or labour turnover.

Business question	Independent variable (x)	Dependent variable (y)
Does advertising increase sales?	Advertising spending per month	Sales revenue per month
How does price affect demand?	Price	Quantity sold
Does training raise productivity?	Training hours per employee	Output per worker
Do ice cream sales depend on the weather?	Daily maximum temperature	Ice creams sold per day
Do older machines produce more faults?	Age of machine (years)	Defect rate (%)
Do retail sales follow household income?	Disposable income	Retail sales

Scatter diagrams

A scatter diagram plots each pair of observations as a single point: one week’s advertising spending and that week’s sales, or one machine’s age and its defect rate. The pattern of the points shows at a glance whether the two variables are related, in which direction and how closely.

Decide which variable is independent (x-axis) and which is dependent (y-axis).
Choose scales that fit all the data, with equal intervals on each axis; an axis may start above zero if the break is clearly shown.
Label each axis with the variable and its units, and give the diagram a title.
Plot each pair of values as a small cross or dot. Do not join the points.
Describe the pattern, and add a line of best fit if the points lie roughly along a straight line.

The data for the worked example come from an invented business. PowerHouse Gyms records its advertising spending and the number of new members who join in six months (hypothetical data).

Month	1	2	3	4	5	6
Advertising spending, x ($000)	2	3	5	6	8	10
New members, y	38	50	62	74	88	108

Correlation

Correlation describes how closely two variables move together. It has a direction (positive or negative) and a strength (strong, moderate or weak). When there is no pattern in the points, there is no correlation.

Positive correlation: as x increases, y tends to increase (advertising spending and sales; temperature and sales of cold drinks).
Negative correlation: as x increases, y tends to decrease (price and quantity demanded; age of a machine and its output per hour).
No correlation: there is no linear relationship (the height of employees and their sales performance).

Strength is judged by how tightly the points cluster around a straight line. In a strong correlation the points lie close to the line and a prediction from it is fairly reliable; in a weak correlation they are widely scattered and predictions are uncertain. A non-linear pattern, such as sales that rise quickly at first and then level off, may show a clear relationship even though a straight line fits it poorly.

The correlation coefficient

The correlation coefficient (Pearson’s r) puts a number on the direction and strength of a linear relationship. It ranges from −1 to +1: +1 is a perfect positive correlation, with every point exactly on a rising line; −1 a perfect negative correlation; and 0 no linear correlation. It is calculated by statistics software, spreadsheets or a calculator.

Value of r	Description
+0.8 to +1	Strong positive correlation
+0.4 to +0.8	Moderate positive correlation
0 to +0.4	Weak positive correlation
0	No linear correlation
−0.4 to 0	Weak negative correlation
−0.8 to −0.4	Moderate negative correlation
−1 to −0.8	Strong negative correlation

The boundaries in the table are conventions, and textbooks use slightly different cut-offs. The PowerHouse Gyms data give r = 0.997, a very strong positive correlation.

Optional depth: what does r² mean?

Squaring the correlation coefficient gives r², the coefficient of determination. It shows the proportion of the variation in the dependent variable that is explained by the straight-line relationship with the independent variable. If r = 0.9, then r² = 0.81, so 81% of the variation in y is associated with variation in x and 19% is due to other factors.

Real business data are rarely as neat as the gym example. Annual figures for the United States from 2000 to 2024 show a weak negative correlation between the unemployment rate and consumer sentiment (an index of how confident households feel about their finances and the economy): r = −0.39.

Scatter diagram of US consumer sentiment index against the unemployment rate, annual averages 2000 to 2024, with the downward line y = 97.98 − 2.65x and r = −0.39. Points are widely scattered; 2022 has low unemployment (3.6%) and very low sentiment (59). — Sentiment tends to be lower when unemployment is higher, but the points are widely scattered: in 2022 unemployment was only 3.6% yet sentiment was at its lowest. Source: US BLS (UNRATE) and University of Michigan (UMCSENT) via FRED, 2026.

The line slopes downwards, as the theory suggests, but many years lie far from it. Sentiment depends on other factors as well as unemployment; 2022, for example, was a year of high inflation, when consumer prices rose by about 8%. With r = −0.39, r² is only about 0.15, so unemployment is associated with about 15% of the variation in sentiment, and a retailer using unemployment alone to predict consumer confidence would often be badly wrong.

Sources: US Bureau of Labor Statistics, Unemployment Rate (UNRATE) and Consumer Price Index (CPIAUCSL); University of Michigan, Consumer Sentiment (UMCSENT); via FRED, retrieved October 2026; annual averages of monthly data.

The line of best fit

A line of best fit summarises the relationship shown by a scatter diagram with a single straight line. It can be drawn by eye or calculated by the least-squares method, and it is the basis of any prediction made from the data.

Drawing a line of best fit by eye

Calculate the mean of the x-values and the mean of the y-values, and plot the mean point; the line should pass through it.
Draw a single straight line through the mean point with roughly equal numbers of points above and below it along its length.
Ignore obvious outliers when judging the line, but mention them in the analysis.
Do not force the line through the origin or through the first and last points.

A line drawn by eye is quick and good enough for many decisions, but two people may draw slightly different lines through the same points. The least-squares method gives the one line that fits best by a precise rule.

The least-squares line: y = a + bx

Every straight line can be written as y = a + bx. The constant a is the intercept, the value of y when x is zero, where the line crosses the y-axis. The coefficient b is the gradient, or slope: the change in y for each one-unit increase in x. A positive b gives a rising line (positive correlation) and a negative b a falling line.

For each point, the vertical distance between the point and a line is its residual: the error the line would make in predicting that point. The least-squares method chooses a and b so that the sum of the squared residuals is as small as possible. Squaring makes all the errors positive and gives more weight to large errors.

b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - \left(\sum x\right)^2} \qquad a = \bar{y} - b\bar{x}

Here n is the number of pairs of data, $ar{x}$ and $ar{y}$ are the means of x and y, and Σ means “the sum of”. The second formula shows that the least-squares line always passes through the mean point $(ar{x}, ar{y})$ .

Using the PowerHouse Gyms data, calculate the equation of the least-squares line of best fit, and explain the meaning of a and b.

Show the solution

Step 1: tabulate x, y, xy and x².

x ($000)	y (members)	xy	x²
2	38	76	4
3	50	150	9
5	62	310	25
6	74	444	36
8	88	704	64
10	108	1,080	100
Σx = 34	Σy = 420	Σxy = 2,764	Σx² = 238

Step 2: calculate b with n = 6.

b = \frac{6 \times 2{,}764 - 34 \times 420}{6 \times 238 - 34^2} = \frac{16{,}584 - 14{,}280}{1{,}428 - 1{,}156} = \frac{2{,}304}{272} = 8.47

Using the line: interpolation and extrapolation

Once the line is known, it can be used to estimate y for any value of x, by reading from the graph or by substituting x into the equation. The reliability of the estimate depends on whether x lies inside or outside the range of the data used to draw the line.

Interpolation is fairly safe when the correlation is strong, because the line has been tested by data on both sides of the estimate. Extrapolation extends the line beyond the data, and assumes that the relationship continues unchanged into a range where it has never been observed. Extrapolation is also the basis of trend forecasting in time series analysis, where time is the independent variable.

Regression line y = 22.0 + 8.47x of new gym members against advertising spending. Six points lie between 2 and 10 thousand dollars, a shaded band labelled interpolation; at 7 a guide reads 81 members. Beyond 10 the dashed line is labelled extrapolation and reaches 191? at 20. — Within the data (shaded) the line gives about 81 members for $7,000 of advertising; extended to $20,000 it predicts 191, but nothing in the data shows the relationship holds that far. Hypothetical data.

Using y = 22.0 + 8.47x, calculate the expected number of new members for advertising spending of (a) $7,000 and (b) $20,000 a month, and comment on the reliability of each estimate.

Show the solution

(a) y = 22.0 + 8.47 × 7 = 22.0 + 59.3 = 81.3, about 81 new members. This is interpolation: $7,000 lies within the range of the data ($2,000–$10,000) and r = 0.997, so the estimate is fairly reliable.

(b) y = 22.0 + 8.47 × 20 = 22.0 + 169.4 = 191.4, about 191 new members. This is extrapolation: $20,000 is twice the highest spending observed. The estimate assumes each extra $1,000 keeps bringing 8.5 members, but advertising usually shows diminishing returns, as the same potential customers see the advertisements repeatedly, and the gym may not have the capacity for so many new members. The figure is unreliable.

The relationship may change outside the range: diminishing returns to advertising, capacity limits, saturation of the market.
Other factors may change: a competitor opens, incomes fall, tastes shift.
The further the extrapolation, the greater the risk, because small errors in the gradient grow with distance.

Correlation is not causation

A strong correlation shows that two variables move together. It does not show that one causes the other. Causation means that a change in one variable directly brings about a change in the other, and proving it needs more than a scatter diagram: a sound business explanation, evidence that the cause comes before the effect, and ideally a controlled test.

There are three common reasons why two variables can be correlated without one causing the other:

A third (confounding) factor affects both. Ice cream sales and sunburn cases rise together, but both are caused by hot weather; banning ice cream would not reduce sunburn.
Reverse causation: the direction runs the other way. A business may find that its most profitable branches spend most on staff training, but the profit may be what pays for the training rather than the result of it.
Coincidence: two series that both rise over time, such as a firm’s sales and the number of smartphone users in the world, will be correlated even if they are unrelated. Trending time series often show high correlations for this reason.

CrashCourse Statistics: why correlation does not prove causation, with examples of third factors, reverse causation and coincidental (spurious) correlations.

Businesses test causation by experiment where they can. An online retailer can show one version of a web page to half its visitors and another to the other half (an A/B test); because the groups are chosen at random, a difference in sales between them can be attributed to the page. Where experiments are impossible, managers combine the statistical evidence with market research and judgment.

Uses of simple linear regression in business

Area	Independent variable (x)	Dependent variable (y)	Decision informed
Marketing	Advertising spending; price	Sales; quantity sold	Size of the marketing budget; pricing
Sales forecasting	Time (years, quarters); household income	Sales revenue	Production, stock and cash flow plans
Human resources	Training hours; pay relative to competitors	Productivity; labour turnover	Training budgets; pay policy
Operations	Age of machinery; output	Defect rate; total cost	Replacing machines; estimating costs at new output levels
Finance	Sales volume	Semi-variable costs	Budgets; separating fixed and variable elements of a cost

In sales forecasting, regression provides a trend line for a time series and a way to forecast sales from variables that are themselves forecast, such as incomes or population. In operations, regressing total cost on output separates a semi-variable cost into a fixed part (the intercept a) and a variable cost per unit (the gradient b), which feeds into break-even analysis.

Regression results are easy to calculate with a spreadsheet, so the main risk lies in interpreting them. A marketing manager who finds that sales rose with advertising spending must still ask whether something else changed at the same time, such as a seasonal peak or a competitor’s price rise.

Evaluation: strengths and limitations of simple linear regression

Regression turns a vague impression (“advertising seems to work”) into a measured relationship (“each $1,000 is associated with about 8.5 new members”) that can be used to plan budgets and forecasts. It is objective, can be repeated as new data arrive, and the strength of the relationship can be checked with r.

Its main limitations are these:

Correlation is not causation. The line measures association; it does not explain it.
Only one independent variable. Sales depend on price, advertising, competitors, incomes and the season at the same time. Simple regression considers one, so its results may reflect the others.
Only straight lines. Many business relationships are curved (diminishing returns to advertising, saturation of a market), and a straight line misrepresents them.
Sensitive to outliers and to the data chosen. One unusual month can tilt the line, and choosing a different period can change the result.
The past may not hold in the future. Extrapolation assumes the relationship is stable, which a change in the market can break.
Data quality and quantity. A handful of points gives an unreliable line, however high r appears.

How useful regression is therefore depends on the strength and stability of the relationship, the number and quality of observations, whether there is a sound business reason for the link, and how far the prediction lies from the data. It is most valuable as one input to a decision, alongside market research and managers’ knowledge of the market, rather than as proof that one variable causes another.

Case study: US retail sales and household income

Retailers, shopping centre owners and consumer goods makers watch household income closely, because spending in shops depends on how much money households have left after tax. US data on disposable personal income (income after taxes) and retail sales (excluding restaurants and bars) show how regression can turn that link into a forecasting tool, and how its forecasts can fail.

Year	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Disposable income, x ($ trillion)	11.32	11.86	12.41	12.39	13.00	13.53	13.93	14.61	15.45	16.16
Retail sales, y ($ trillion)	3.74	4.02	4.21	4.38	4.55	4.63	4.74	4.95	5.17	5.33

Sources: US Bureau of Economic Analysis, Disposable Personal Income (DSPI, annual average of seasonally adjusted annual rates); US Census Bureau, Advance Retail Sales: Retail Trade (RSXFS, annual total of seasonally adjusted monthly sales); via FRED, retrieved October 2026; current dollars.

Over 2010–2019 the relationship is very close to a straight line. The least-squares line is y = 0.36 + 0.313x, with a correlation coefficient of r = 0.98 (r² = 0.97). The gradient means that, over the decade, each extra $1 trillion of disposable income was associated with about $313 billion more retail sales, roughly 31 cents of every extra dollar of income.

Scatter diagram of US retail sales against disposable personal income, 2010 to 2019, both in trillions of dollars. Ten points rise from (11.3, 3.7) in 2010 to (16.2, 5.3) in 2019, close to the line y = 0.36 + 0.313x; r = 0.98. — From 2010 to 2019 retail sales rose almost exactly in line with disposable income (r = 0.98). The axes start at $10 trillion and $3 trillion. Source: US BEA (DSPI) and US Census Bureau (RSXFS) via FRED, 2026.

A retailer in 2019 could have used this line to forecast sales from forecasts of income. When the 2010–2019 line is extended to the incomes of 2020–2025, its predictions differ from the actual outcomes, sometimes by a wide margin.

Year	Disposable income ($ trillion)	Predicted retail sales ($ trillion)	Actual retail sales ($ trillion)	Error (actual − predicted)
2020	17.38	5.80	5.50	−0.30
2021	18.81	6.25	6.44	+0.19
2022	18.92	6.28	6.97	+0.69
2023	20.81	6.87	7.03	+0.16
2024	22.05	7.26	7.09	−0.17
2025	22.99	7.56	7.32	−0.24

The 2010 to 2019 regression line of US retail sales on disposable income extended as a dashed line to an income of 24 trillion dollars. Actual 2020 to 2025 points are hollow circles: 2020 below the line, 2021 to 2023 above it (2022 well above), 2024 and 2025 slightly below. — The dashed line extrapolates the 2010–2019 relationship. Sales fell below it in 2020, rose well above it in 2021–2022 and fell back below it in 2024–2025. Source: US BEA and US Census Bureau via FRED, 2026.

Each error has an explanation that the line could not include. In 2020 incomes rose because of government support payments during the pandemic, but many shops were closed and households saved more (the personal saving rate reached about 32% of disposable income in April 2020), so sales were $0.30 trillion below the prediction.

In 2021–2022 spending rebounded and prices rose quickly (consumer prices rose about 8% in 2022), so nominal sales in 2022 were $0.69 trillion, about 10%, above the prediction. By 2024–2025 sales were below the line again; the retail figures exclude spending on services such as travel and restaurants, so any income spent on them does not appear in the data.

The case also shows why correlation is not causation. Both series are in current dollars, so inflation and population growth push both upwards whatever the link between them; part of the very high r reflects these shared trends. Income does influence spending, but the strength of the correlation overstates how precisely one can be predicted from the other. A retailer would use the regression as a starting point and adjust it with knowledge of prices, government policy and consumer behaviour.

Sources: US Bureau of Economic Analysis (DSPI); US Census Bureau (RSXFS); US Bureau of Labor Statistics (CPIAUCSL); US BEA, Personal Saving Rate (PSAVERT); via FRED, retrieved October 2026. Regression calculated from the rounded annual figures shown.

Explain what the gradient of 0.313 shows about US retail sales and disposable income in 2010–2019.
Using the line y = 0.36 + 0.313x, calculate the predicted retail sales for an income of $15 trillion, and state whether this is interpolation or extrapolation.
Discuss whether a US retailer should use this regression to forecast its sales for the next three years.

Exam practice

Simple linear regression appears in HL Paper 2, where you may be given paired data or a scatter diagram, asked to plot points or draw a line of best fit, to describe the correlation, to use a given equation to estimate a value, and to comment on the reliability of the estimate. It can also support analysis in Paper 3 or the internal assessment when the data allow it.

Short questions often give the equation of the line and ask for an estimate and a comment.

Cool Cabana, a beach kiosk in Goa, finds that over 40 days its daily sales of cold drinks (y) are related to the maximum temperature in °C (x) by the line y = −120 + 14x, with r = 0.85 (hypothetical data). Temperatures on those days ranged from 26 °C to 36 °C.

(a) Calculate the expected sales on a day with a maximum of 32 °C. [2 marks]

(b) Comment on the reliability of using the line to predict sales on a day of 42 °C. [2 marks]

Show the solution

(a) y = −120 + 14 × 32 = −120 + 448 = 328 cold drinks.

(b) 42 °C lies outside the observed range of 26–36 °C, so the estimate (−120 + 14 × 42 = 468 drinks) is an extrapolation. The correlation is strong (r = 0.85), but at extreme temperatures fewer people may visit the beach, and the kiosk may run out of stock or staff, so the relationship may not hold; the estimate is unreliable.

[10 marks] Saber Mais is an online tutoring company in Brazil. Its marketing director has plotted monthly spending on social media advertising against new subscriptions for the past 18 months and found a strong positive correlation (r = 0.82), with the line y = 150 + 0.9x (y = new subscriptions, x = advertising in BRL 000). Advertising spending ranged from BRL 40,000 to BRL 120,000 a month. She proposes to raise advertising to BRL 300,000 a month during the next school year, arguing that the line predicts 420 new subscriptions a month. Discuss whether Saber Mais should base its advertising budget on this regression.

Show an answer plan

Plan.

Introduction: define simple linear regression, correlation and extrapolation; explain y = 150 + 0.9x (b = 0.9 new subscriptions for each extra BRL 1,000).
Arguments for: 18 observations and r = 0.82 show a strong positive correlation; the gradient gives a measurable return on advertising that can be compared with the revenue per subscriber; the method is objective and can be updated monthly.
Arguments against: BRL 300,000 is 2.5 times the highest spending observed, so the forecast is an extrapolation; diminishing returns as the same audience sees repeated advertisements; correlation is not causation, since a third factor (the school calendar: both advertising and subscriptions may rise before exams) could explain the link; only one independent variable (prices, competitors and word of mouth are ignored); 18 months is a short period.
Other evidence: an A/B test of higher spending in some regions or platforms; cost per new subscriber compared with lifetime revenue; market research on why students subscribe.
Conclusion: use the regression as evidence that advertising works within the range observed, but raise spending in steps (for example to BRL 150,000) while monitoring the cost per subscription, rather than committing to BRL 300,000 on an extrapolated line. Limitations of the case: no data on revenue per subscriber or on seasonality.

Summary

Simple linear regression measures the relationship between an independent variable (x-axis) and a dependent variable (y-axis) and fits a straight line, y = a + bx.
A scatter diagram plots paired data as points; correlation describes its direction (positive, negative, none) and strength (strong to weak).
The correlation coefficient r runs from −1 to +1; values close to ±1 show a strong linear relationship.
The least-squares line minimises the sum of squared vertical distances; b is the change in y per unit of x and a is the intercept.
Interpolation estimates within the range of the data and is fairly reliable; extrapolation goes beyond it and is risky.
Correlation is not causation: a third factor, reverse causation or coincidence may explain a correlation.
Regression supports forecasting and decisions in every function but works best combined with market research and judgment.

Self-check

To do:
I can identify the independent and dependent variables in a business relationship and put them on the correct axes.
To do:
I can construct a scatter diagram from paired data.
To do:
I can describe the direction and strength of a correlation from a scatter diagram or a value of r.
To do:
I can draw a line of best fit by eye through the mean point.
To do:
I can calculate the least-squares line y = a + bx for a small data set and explain a and b.
To do:
I can use a line of best fit to interpolate and extrapolate, and explain why extrapolation is riskier.
To do:
I can explain why correlation does not prove causation, with an example of a third factor, reverse causation or coincidence.
To do:
I can apply simple linear regression to a business decision and evaluate its reliability.

Key terms introduced here

Carry on with T.15 in Augment

Key terms introduced here

Carry on with T.15 in Augment

Notes on T.15 Simple linear regression

Learning objectives

Introduction

Independent and dependent variables

Scatter diagrams

Correlation

The correlation coefficient

The line of best fit

Drawing a line of best fit by eye

The least-squares line: y = a + bx

Using the line: interpolation and extrapolation

Correlation is not causation

Uses of simple linear regression in business

Evaluation: strengths and limitations of simple linear regression

Case study: US retail sales and household income

Exam practice

Summary

Self-check

Flashcards for T.15