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CFA Level I Exam · Applications of Simple Linear Regression in Finance

ANOVA, R-squared and Standard Error of Estimate Explained

Updated 7 October 2026 · Fact-checked

The ANOVA table splits total variation in Y (SST) into explained variation (SSR) and unexplained variation (SSE). R-squared = SSR ÷ SST. The standard error of estimate = √(SSE ÷ (n − 2)). The F-statistic = MSR ÷ MSE, with 1 and n − 2 degrees of freedom in simple regression.

Understand ANOVA, R-squared and Standard Error of Estimate

A regression line never fits the data perfectly. Your job is to measure how well it fits. The ANOVA table does this by breaking the variation in the dependent variable Y into two parts.

SST (total sum of squares) is the total variation of Y around its mean: Σ(Yi − Ȳ)². SSR (regression sum of squares) is the part explained by the regression: Σ(Ŷi − Ȳ)². SSE (sum of squared errors, also called residual sum of squares) is the part left unexplained: Σ(Yi − Ŷi)². The link is simple: SST = SSR + SSE.

The coefficient of determination, R², is the share of total variation that the regression explains: R² = SSR ÷ SST = 1 − SSE ÷ SST. In simple linear regression with one independent variable, R² also equals the square of the correlation between X and Y. So r = ±√R², and the sign of r matches the sign of the slope.

R² is unitless, but the standard error of estimate (SEE) is in the units of Y. It measures the typical size of the residuals. A lower SEE means a tighter fit. SEE = √MSE, where MSE = SSE ÷ (n − 2). We divide by n − 2 because two parameters (intercept and slope) were estimated.

The F-test asks whether the slope is zero. F = MSR ÷ MSE, where MSR = SSR ÷ k and k = 1 in simple regression. Degrees of freedom are 1 (numerator) and n − 2 (denominator). Here F equals the square of the slope's t-statistic, so both tests give the same conclusion. The F-test is one-tailed: you reject H0 (slope = 0) only when F is large.

Key formulas to remember

Sum of squares decomposition
SST = SSR + SSE
SST = Σ(Yi − Ȳ)²; SSR = Σ(Ŷi − Ȳ)²; SSE = Σ(Yi − Ŷi)².
Coefficient of determination
R² = SSR ÷ SST = 1 − SSE ÷ SST
Lies between 0 and 1. In simple regression, R² = r².
Mean square regression
MSR = SSR ÷ k, with k = 1
k is the number of independent variables.
Mean square error
MSE = SSE ÷ (n − 2)
Degrees of freedom are n − 2 in simple regression.
Standard error of estimate
SEE = √MSE = √[SSE ÷ (n − 2)]
In the units of the dependent variable.
F-statistic
F = MSR ÷ MSE, df = 1 and n − 2
Tests H0: slope = 0. One-tailed, reject if F is above the critical value. F = t² for the slope.
Correlation from R²
r = ±√R²
Take the sign of the slope coefficient.

How to solve ANOVA, R-squared and Standard Error of Estimate questions

Use this routine for any ANOVA, R² or SEE question. First find out which pieces the question gives you, then fill in the rest of the table.

  1. 1Write the table skeleton: SSR (df = 1), SSE (df = n − 2), SST (df = n − 1).
  2. 2Fill what you know, then use SST = SSR + SSE to find a missing sum of squares.
  3. 3Compute the mean squares: MSR = SSR ÷ 1 and MSE = SSE ÷ (n − 2).
  4. 4Compute R² = SSR ÷ SST, or r = ±√R² if correlation is asked. Use the slope's sign.
  5. 5Compute SEE = √MSE. Do not forget the square root.
  6. 6Compute F = MSR ÷ MSE and compare it with the critical F value with 1 and n − 2 df, if a test is asked.
  7. 7Check that your answer is sensible: R² is between 0 and 1, and SEE is in the units of Y.

Quickest way: Three-ratio shortcut

When to use it: When the question gives two of SSR, SSE and SST, or gives R² and n.

  1. Get the missing sum of squares from SST = SSR + SSE.
  2. R² = SSR ÷ SST. Then F = R² ÷ (1 − R²) × (n − 2) in simple regression.
  3. SEE = √[SSE ÷ (n − 2)]. Use SSE = SST × (1 − R²) if only R² is given.
  4. Eliminate options that break the rules: R² above 1, SEE below zero, or the wrong df.

Common mistakes in ANOVA, R-squared and Standard Error of Estimate

  • Dividing SSE by n instead of n − 2 when computing SEE.

    It looks like an ordinary variance formula, where you divide by n or n − 1.

    Fix: Two parameters are estimated in simple regression, so always use n − 2.

  • Forgetting the square root and reporting MSE as the SEE.

    Students stop once they have SSE ÷ (n − 2).

    Fix: SEE = √MSE. Check that the units match Y, not Y squared.

  • Using SSE ÷ SST as R².

    Students mix up explained and unexplained variation.

    Fix: R² uses SSR (explained) in the numerator. SSE ÷ SST is the unexplained share, 1 − R².

  • Taking r = +√R² when the slope is negative.

    A square root has two signs and students ignore this.

    Fix: Give r the same sign as the slope coefficient.

  • Using n − 1 or n as the denominator degrees of freedom for F.

    Students confuse the regression df with total df.

    Fix: In simple regression, F has 1 and n − 2 degrees of freedom. SST has n − 1.

  • Reading a high R² as proof the model is correct or that X causes Y.

    R² feels like a score of quality.

    Fix: R² only measures the share of variation explained. It says nothing about causation or whether the assumptions hold.

Worked examples

Example 1

A simple linear regression of a stock's monthly return on a market index uses 32 observations. The ANOVA shows SSR = 180 and SSE = 120. Compute R² and the standard error of estimate. Options for SEE: A) 1.83, B) 2.00, C) 2.45.

Show the solution
  1. SST = SSR + SSE = 180 + 120 = 300.
  2. R² = SSR ÷ SST = 180 ÷ 300 = 0.60.
  3. Degrees of freedom for SSE = n − 2 = 30.
  4. MSE = 120 ÷ 30 = 4.0.
  5. SEE = √4.0 = 2.00.

Answer: R² = 0.60 and SEE = 2.00 (option B).

Example 2

A regression of Y on X with 27 observations has R² = 0.40 and SST = 150. The slope is negative. Find the F-statistic and the correlation between X and Y. Options for F: A) 10.00, B) 16.67, C) 24.00.

Show the solution
  1. SSR = R² × SST = 0.40 × 150 = 60.
  2. SSE = SST − SSR = 150 − 60 = 90.
  3. MSR = 60 ÷ 1 = 60.
  4. MSE = 90 ÷ (27 − 2) = 90 ÷ 25 = 3.6.
  5. F = 60 ÷ 3.6 = 16.67, with 1 and 25 df.
  6. r = −√0.40 = −0.632, negative because the slope is negative.

Answer: F = 16.67 (option B) and r ≈ −0.632.

Exam tips

  • If SST, SSR or SSE is missing, always rebuild it from SST = SSR + SSE first. Many items are just this one step.
  • Watch for wrong-answer traps: SSE ÷ SST used as R², n used instead of n − 2, and MSE reported without the square root.
  • The F-test in simple regression gives the same result as the two-tailed t-test on the slope, because F = t². You can use either one.
  • A larger SEE means a poorer fit. If a question compares two models, the lower SEE and higher R² fit better, assuming the same dependent variable.
  • Use your calculator's √ key and check the order: divide first, then take the square root.

Practice questions from Applications of Simple Linear Regression in Finance

ANOVA, R-squared and Standard Error of Estimate in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

ANOVA, R-squared and Standard Error of Estimate: frequently asked questions

What is the difference between SSE, SSR and SST?

SST is the total variation of Y around its mean. SSR is the part of it the regression explains, and SSE is the unexplained part. They satisfy SST = SSR + SSE.

How do I calculate R-squared from SSR and SST?

Divide SSR by SST. For example, SSR = 180 and SST = 300 give R² = 0.60. You can also compute 1 − SSE ÷ SST and get the same answer.

What is the standard error of estimate formula in CFA Level I?

SEE = √[SSE ÷ (n − 2)] for simple linear regression. It is the square root of the mean square error. It is measured in the units of the dependent variable.

What does the F-test tell you in simple linear regression?

It tests whether the slope is zero, using F = MSR ÷ MSE with 1 and n − 2 degrees of freedom. A large F leads you to reject H0. In simple regression F equals the slope's t-statistic squared.