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By Daniil Tarakanov Updated Oct 16, 20262026 curriculum

The ANOVA table and the standard error of estimate

Analysis of variance (ANOVA) splits the total variation in Y, the sum of squares total (SST), into the part the regression explains, the sum of squares regression (SSR), and the part it does not, the sum of squares error (SSE). Dividing each by its degrees of freedom gives mean squares. F = MSR ÷ MSE (mean square regression over mean square error) tests the model, and the standard error of estimate, SEE = √MSE, is the typical size of a residual.

6 min cram, 30 min full read. Based on the 2026 CFA® Program Level I curriculum, Quantitative Methods, Learning Module 10. This is the 2026 version of the lesson. Sitting in February 2027 or later? Read the updated 2027 version.

Tailor this to you: take the self-assessment

Before this: Goodness of fit and testing the regression coefficients

On the exam

Describe the use of analysis of variance (ANOVA) in regression analysis, interpret ANOVA results, and calculate and interpret the standard error of estimate in a simple linear regression

Describe ANOVA in regression, interpret an ANOVA table, and calculate the standard error of estimate. Expect a table with blanks to fill from any three of SSR, SSE, SST and n. Wrong options divide SSE by n instead of n − 2, forget the square root, or use sums of squares in F.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

What does ANOVA do in a regression?

Analysis of variance (ANOVA) splits the variation in Y into the part the regression explains and the part it does not, then compares them. The regression output tabulates the pieces.

Start with the three sums of squares. The sum of squares total (SST) measures how far the Y values are from their mean. The sum of squares regression (SSR) measures how far the fitted values are from that mean (the explained part). The sum of squares error (SSE) measures how far the actual values are from the fitted line (the unexplained part).

Decomposition of variation Memorize

SST = SSR + SSE

What does the table look like?

For the six retailers from the previous lessons:

Source	Sum of squares	df	Mean square	F
Regression	SSR = 62.727	1	MSR = 62.727	90.47
Error	SSE = 2.773	n − 2 = 4	MSE = 0.693
Total	SST = 65.500	n − 1 = 5

Each mean square is a sum of squares divided by its degrees of freedom. The error degrees of freedom are $n-2$ because two coefficients (intercept and slope) were estimated from the data. The regression has 1 degree of freedom because there is one slope.

Mean square error Memorize

MSE = \frac{SSE}{n-2}

The mean square error is the average squared miss. Its square root is the typical size of a miss, the standard error of estimate (SEE).

Standard error of estimate Memorize

SEE = \sqrt{MSE} = \sqrt{\frac{SSE}{n-2}}

For the retailers, $SEE = \sqrt{0.6933} = 0.8327$ percentage points: a typical miss of about 0.8 points of revenue growth. SEE is in the units of Y. $R^2$ has no units.

Rearrange itStandard error of estimate: solve for any one

The question asks for

A simple regression on 6 observations has SSE = 2.7733. The standard error of estimate is closest to:

Sum of squares error, SSEObservations, n

S E E = \frac{S S E}{n - 2}

√(2.7733 ÷ 4) = 0.8327

Standard error of estimate, SEE0.8327

On your calculator

2.7733÷(6−2)=√x

√x square root

2.7733ENTER6ENTER2−÷g√x

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

How is F built from the table?

The F-statistic is the ratio of the two mean squares. It uses mean squares, never the raw sums.

F-statistic Memorize

F = \frac{MSR}{MSE} = \frac{SSR \,/\, 1}{SSE \,/\, (n-2)}

Where each ANOVA number comes from

Total variation, SSTdf = n − 1

Explained by the line: SSR Memorize df = 1 in simple regression
- MSR = SSR ÷ 1 the numerator of F
- R² = SSR ÷ SST share explained
Left unexplained: SSE Memorize df = n − 2
- MSE = SSE ÷ (n − 2) the denominator of F
- SEE = √MSE typical size of a residual, in the units of Y

InteractiveComplete the ANOVA table

Try this 0 of 3

Fill in MSE, F, R² and SEE for the worked example
Solve a new puzzle with every cell right (no revealing)
Solve a puzzle that gives R² or SEE instead of a second sum of squares

Worked example: six retailers (n = 6). Fill in MSE, F, R² and SEE.

ANOVA table: type each missing value. F, R² and SEE are below the table.
Source	SS	df	MS
Regression	62.727	1	62.727
Error	2.773	4
Total	65.5	5

F =R² =SEE =

Cells right0 of 4

Observations n6

GivenSS, df and MSR

Type a value in any blank cell to check it. Each answer shows the relation it comes from.

Show working

n = 6. Degrees of freedom: regression 1, error n − 2 = 4, total n − 1 = 5.

SST = SSR + SSE: 62.727 + 2.773 = 65.5 (given)

SSR = SST − SSE: 65.5 − 2.773 = 62.727 (given)

SSE: 2.773 (given)

MSR = SSR ÷ 1: 62.727 ÷ 1 = 62.727 (given)

MSE = SSE ÷ (n − 2): 2.773 ÷ 4 = 0.693

F = MSR ÷ MSE: 62.727 ÷ 0.693 = 90.47

R² = SSR ÷ SST: 62.727 ÷ 65.5 = 0.9577

SEE = √MSE: √0.693 = 0.833

SEE = 0.833 is the typical size of a residual, in Y's units. R² = 0.958 is the share of SST the regression explains, with no units.

MSE, F, SEE and R² on your calculator

2.7733÷4=STO1MSE = 0.6933
62.7267÷RCL1=F = 90.47
RCL1√xSEE = 0.8327
62.7267÷65.5=R² = 0.9577

STO store · RCL recall a stored number · √x square root

2.7733ENTER4÷STO1MSE = 0.6933
62.7267RCL1÷F = 90.47
RCL1g√xSEE = 0.8327
62.7267ENTER65.5÷R² = 0.9577

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number · g blue shift key · √x square root

Display: MSE 0.6933, F 90.47, SEE 0.8327, R² 0.9577

MSE divides by n − 2, never n. SEE needs the square root. F uses mean squares, not sums of squares.

Three numbers and n unlock the whole table.

How do you complete a table with blanks?

The exam gives you any three of SSR, SSE, SST and $n$ , and you fill in the rest. Use $SST = SSR + SSE$ to get the missing sum of squares, the degrees of freedom $1$ , $n-2$ and $n-1$ , then divide.

Worked example Fill in the blanks

A regression has $n = 22$ , $SSR = 90$ and $SSE = 60$ .

$SST = 90 + 60 = 150$ . Degrees of freedom: 1, 20 and 21.
$MSR = 90 \div 1 = 90$ and $MSE = 60 \div 20 = 3.00$ .
$F = 90 \div 3.00 = 30.0$ .
$SEE = \sqrt{3.00} = 1.732$ , and $R^2 = 90 \div 150 = 0.60$ .

Steps 2 to 4 on your calculator

60÷20=STO1 MSE 3
90÷RCL1= F 30
RCL1√x SEE 1.7321

STO store · RCL recall a stored number · √x square root

60ENTER20÷STO1 MSE 3
90RCL1÷ F 30
RCL1g√x SEE 1.7321

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number · g blue shift key · √x square root

Rearrange itF-statistic from the ANOVA table: solve for any one

The question asks for

A simple regression on 6 observations has SSR = 62.727 and SSE = 2.7733. The F-statistic is closest to:

Sum of squares regression, SSRSum of squares error, SSEObservations, n

F = \frac{S S R / 1}{S S E / ( n - 2 )}

62.727 ÷ (2.7733 ÷ 4) = 90.4727

F-statistic90.4727

On your calculator

62.727÷(2.7733÷(6−2))=

62.727ENTER2.7733ENTER6ENTER2−÷÷

ENTER enter (on the HP, puts the number on the stack)

Exam tip Check your table with the degrees of freedom: regression plus error must equal total (1 + 20 = 21). A mismatch means you used n instead of n − 2.

Go deeper Why the error degrees of freedom are n − 2

The residuals are computed from a line whose two coefficients were chosen using the same data. Those two choices tie down two of the $n$ residuals (they sum to zero, and they are uncorrelated with X), so only $n-2$ are free to vary. Dividing SSE by $n-2$ , not $n$ , corrects for that and gives an unbiased estimate of the error variance.

Go deeper SEE versus the standard deviation of Y

The standard deviation of Y is the spread around its mean, $\sqrt{SST \div (n-1)}$ . The SEE is the spread around the regression line, $\sqrt{SSE \div (n-2)}$ . When X explains a lot, SEE is much smaller than the standard deviation of Y. For the retailers, the standard deviation of Y is 3.62 and the SEE is 0.83.

Memorize this

MemorizeThe ANOVA table

Source	Sum of squares	df	Mean square	F
Regression	SSR	$1$	$MSR = SSR \div 1$	$MSR \div MSE$
Error	SSE	$n-2$	$MSE = SSE \div (n-2)$
Total	SST = SSR + SSE	$n-1$

Memory hook Degrees of freedom add up: 1 + (n − 2) = n − 1. Sums of squares add up too. Mean squares do not.

Know it cold $SEE = \sqrt{MSE}$ (root of SSE ÷ (n − 2)); $R^2 = SSR \div SST$ ; $F = MSR \div MSE$ . Three numbers and n unlock the whole table.

Where each ANOVA number comes from

Total variation, SSTdf = n − 1

Explained by the line: SSR Memorize df = 1 in simple regression
- MSR = SSR ÷ 1 the numerator of F
- R² = SSR ÷ SST share explained
Left unexplained: SSE Memorize df = n − 2
- MSE = SSE ÷ (n − 2) the denominator of F
- SEE = √MSE typical size of a residual, in the units of Y

Decomposition of variation

SST = SSR + SSE

Quick example

SSR = 36 and SSE = 50 give SST = 86. Degrees of freedom split the same way: n − 1 = 1 + (n − 2).

Mean square error

MSE = \dfrac{SSE}{n-2}

Quick example

n = 27 and SSE = 50. MSE = 50 ÷ 25 = 2.00.

On your calculator

50÷(27−2)= 2

50ENTER27ENTER2−÷ 2

ENTER enter (on the HP, puts the number on the stack)

Standard error of estimate

SEE = \sqrt{MSE} = \sqrt{\dfrac{SSE}{n-2}}

Quick example

MSE = 2.00, so SEE = √2.00 = 1.41, in the units of Y.

On your calculator

50÷(27−2)=√x 1.4142

√x square root

50ENTER27ENTER2−÷g√x 1.4142

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

F-statistic

F = \dfrac{MSR}{MSE} = \dfrac{SSR \,/\, 1}{SSE \,/\, (n-2)}

Quick example

SSR = 36, SSE = 50, n = 27. MSR = 36, MSE = 2.00, so F = 36 ÷ 2.00 = 18.0.

On your calculator

36÷(50÷25)= 18

36ENTER50ENTER25÷÷ 18

ENTER enter (on the HP, puts the number on the stack)

How it is asked

SEE from sums of squares

n = 32, SSR = 48, SSE = 72. Standard error of estimate?

MSE = 72 ÷ 30 = 2.40, SEE = √2.40 = 1.55. Traps: √(72 ÷ 32) = 1.50; MSE itself = 2.40.
F from the table

Same regression: F?

F = MSR ÷ MSE = 48 ÷ 2.40 = 20.0. Trap: SSR ÷ SSE = 0.67.
Complete the table

Given n = 27, SSR = 36, SSE = 50, fill in the blanks.

SST 86; df 1, 25, 26; MSE 2.00; F 18.0; SEE 1.41; R² 0.42.
Degrees of freedom

n = 52. Degrees of freedom for SSE and SST?

50 and 51 (n − 2 and n − 1).

Exam traps

Dividing SSE by n (or n − 1) instead of n − 2 to get MSE.
Reporting MSE as the standard error of estimate: SEE = √MSE, so take the root.
Computing F from sums of squares (SSR ÷ SSE) instead of mean squares.
Treating a small SEE alone as a good fit: SEE is in the units of Y, so compare it with the spread of Y or use R².

Checkpoint

4 exam-style questions. About 90 seconds each.

1A regression on 32 observations has SSR = 48 and SSE = 72. The standard error of estimate is closest to:
2For the same regression (n = 32, SSR = 48, SSE = 72), the F-statistic is closest to:
3For the same regression, R² is closest to:
4A simple linear regression uses 52 observations. The degrees of freedom for the sum of squares error and the total sum of squares are, respectively:

Key takeaways

SST = SSR + SSE; the degrees of freedom are 1, n − 2 and n − 1.
MSE = SSE ÷ (n − 2); SEE = √MSE.
F = MSR ÷ MSE; R² = SSR ÷ SST.

Decomposition of variation

SST = SSR + SSE

Quick example

SSR = 36 and SSE = 50 give SST = 86. Degrees of freedom split the same way: n − 1 = 1 + (n − 2).

Mean square error

MSE = \dfrac{SSE}{n-2}

Quick example

n = 27 and SSE = 50. MSE = 50 ÷ 25 = 2.00.

On your calculator

50÷(27−2)= 2

50ENTER27ENTER2−÷ 2

ENTER enter (on the HP, puts the number on the stack)

Standard error of estimate

SEE = \sqrt{MSE} = \sqrt{\dfrac{SSE}{n-2}}

Quick example

MSE = 2.00, so SEE = √2.00 = 1.41, in the units of Y.

On your calculator

50÷(27−2)=√x 1.4142

√x square root

50ENTER27ENTER2−÷g√x 1.4142

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

F-statistic

F = \dfrac{MSR}{MSE} = \dfrac{SSR \,/\, 1}{SSE \,/\, (n-2)}

Quick example

SSR = 36, SSE = 50, n = 27. MSR = 36, MSE = 2.00, so F = 36 ÷ 2.00 = 18.0.

On your calculator

36÷(50÷25)= 18

36ENTER50ENTER25÷÷ 18

ENTER enter (on the HP, puts the number on the stack)

Next lesson Predicting Y and building a prediction interval

Questions about this lesson?

Where this comes back

Predicting Y and building a prediction interval Simple Linear Regression
Simple Linear Regression Quantitative Methods

Questions candidates ask

How do you read an ANOVA table in regression?

The Regression row holds the explained variation (SSR, 1 degree of freedom), the Error row the unexplained (SSE, n − 2) and the Total row their sum (SST, n − 1). Divide each sum of squares by its degrees of freedom for the mean squares, then F = MSR ÷ MSE.

What is the standard error of estimate?

It is the square root of the mean square error, √(SSE ÷ (n − 2)). It measures the typical distance between actual and fitted Y, in the units of Y. A smaller value means the points sit closer to the line.

How do you calculate the F-statistic from an ANOVA table?

Divide the mean square regression by the mean square error: F = MSR ÷ MSE. In simple regression MSR equals SSR because it has one degree of freedom, and MSE is SSE divided by n − 2. A large F means the model explains much more than chance.

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