Randomized Complete Block Design

Probability and Statistics · FE Reference Handbook section

Probability and Statistics

10 formulas

10 exam-style examples

~60 min

All Probability and Statistics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

For k treatments and b blocks
Montgomery, Douglas C., and George C. Runger, Applied Statistics and Probability for Engineers, 4 ed., New York: John Wiley and Sons, 2007.
Engineering Probability and Statistics

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design

A randomized complete block design tests 4 asphalt mixes across 3 blocks. The total sum of squares is 390.0, treatments contribute 160.0 and blocks 35. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 4 treatments, b = 3 blocks$
$SS_total = 390.0$
$SS_treat = 160.0$
$SS_block = 35$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 390.0 - 160.0 - 35 = 195.0$
Degrees of freedom
$df_treat = 3, df_block = 2, df_error = 6$
Mean squares
$MS_treat = 160.0/3 = 53.33; MS_error = 195.0/6 = 32.50$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 53.33/32.50 = 1.641$

Answer:

SS_error = 195.0, F = 1.64 with (3, 6) degrees of freedom

Why the other options are there

F = 0.821 (sums of squares, not mean squares)
df_error = 11 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 2

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (2)

A randomized complete block design tests 4 asphalt mixes across 5 blocks. The total sum of squares is 305.0, treatments contribute 195.0 and blocks 85. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 4 treatments, b = 5 blocks$
$SS_total = 305.0$
$SS_treat = 195.0$
$SS_block = 85$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 305.0 - 195.0 - 85 = 25.0$
Degrees of freedom
$df_treat = 3, df_block = 4, df_error = 12$
Mean squares
$MS_treat = 195.0/3 = 65.00; MS_error = 25.0/12 = 2.08$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 65.00/2.08 = 31.200$

Answer:

SS_error = 25.0, F = 31.20 with (3, 12) degrees of freedom

Why the other options are there

F = 7.800 (sums of squares, not mean squares)
df_error = 19 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 3

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (3)

A randomized complete block design tests 3 asphalt mixes across 3 blocks. The total sum of squares is 590.0, treatments contribute 80 and blocks 70. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 3 treatments, b = 3 blocks$
$SS_total = 590.0$
$SS_treat = 80$
$SS_block = 70$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 590.0 - 80 - 70 = 440.0$
Degrees of freedom
$df_treat = 2, df_block = 2, df_error = 4$
Mean squares
$MS_treat = 80/2 = 40.00; MS_error = 440.0/4 = 110.0$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 40.00/110.0 = 0.364$

Answer:

SS_error = 440.0, F = 0.36 with (2, 4) degrees of freedom

Why the other options are there

F = 0.182 (sums of squares, not mean squares)
df_error = 8 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 4

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (4)

A randomized complete block design tests 4 asphalt mixes across 4 blocks. The total sum of squares is 210.0, treatments contribute 150.0 and blocks 110.0. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 4 treatments, b = 4 blocks$
$SS_total = 210.0$
$SS_treat = 150.0$
$SS_block = 110.0$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 210.0 - 150.0 - 110.0 = -50.0$
Degrees of freedom
$df_treat = 3, df_block = 3, df_error = 9$
Mean squares
$MS_treat = 150.0/3 = 50.00; MS_error = -50.0/9 = -5.56$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 50.00/-5.56 = -9.000$

Answer:

SS_error = -50.0, F = -9.00 with (3, 9) degrees of freedom

Why the other options are there

F = -3.000 (sums of squares, not mean squares)
df_error = 15 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 5

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (5)

A randomized complete block design tests 4 asphalt mixes across 5 blocks. The total sum of squares is 210.0, treatments contribute 200.0 and blocks 40. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 4 treatments, b = 5 blocks$
$SS_total = 210.0$
$SS_treat = 200.0$
$SS_block = 40$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 210.0 - 200.0 - 40 = -30.0$
Degrees of freedom
$df_treat = 3, df_block = 4, df_error = 12$
Mean squares
$MS_treat = 200.0/3 = 66.67; MS_error = -30.0/12 = -2.50$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 66.67/-2.50 = -26.667$

Answer:

SS_error = -30.0, F = -26.67 with (3, 12) degrees of freedom

Why the other options are there

F = -6.667 (sums of squares, not mean squares)
df_error = 19 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 6

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (6)

A randomized complete block design tests 4 asphalt mixes across 4 blocks. The total sum of squares is 245.0, treatments contribute 80 and blocks 45. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 4 treatments, b = 4 blocks$
$SS_total = 245.0$
$SS_treat = 80$
$SS_block = 45$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 245.0 - 80 - 45 = 120.0$
Degrees of freedom
$df_treat = 3, df_block = 3, df_error = 9$
Mean squares
$MS_treat = 80/3 = 26.67; MS_error = 120.0/9 = 13.33$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 26.67/13.33 = 2.000$

Answer:

SS_error = 120.0, F = 2.00 with (3, 9) degrees of freedom

Why the other options are there

F = 0.667 (sums of squares, not mean squares)
df_error = 15 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 7

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (7)

A randomized complete block design tests 3 asphalt mixes across 3 blocks. The total sum of squares is 310.0, treatments contribute 140.0 and blocks 40. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 3 treatments, b = 3 blocks$
$SS_total = 310.0$
$SS_treat = 140.0$
$SS_block = 40$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 310.0 - 140.0 - 40 = 130.0$
Degrees of freedom
$df_treat = 2, df_block = 2, df_error = 4$
Mean squares
$MS_treat = 140.0/2 = 70.00; MS_error = 130.0/4 = 32.50$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 70.00/32.50 = 2.154$

Answer:

SS_error = 130.0, F = 2.15 with (2, 4) degrees of freedom

Why the other options are there

F = 1.077 (sums of squares, not mean squares)
df_error = 8 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 8

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (8)

A randomized complete block design tests 4 asphalt mixes across 3 blocks. The total sum of squares is 465.0, treatments contribute 65 and blocks 65. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 4 treatments, b = 3 blocks$
$SS_total = 465.0$
$SS_treat = 65$
$SS_block = 65$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 465.0 - 65 - 65 = 335.0$
Degrees of freedom
$df_treat = 3, df_block = 2, df_error = 6$
Mean squares
$MS_treat = 65/3 = 21.67; MS_error = 335.0/6 = 55.83$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 21.67/55.83 = 0.388$

Answer:

SS_error = 335.0, F = 0.39 with (3, 6) degrees of freedom

Why the other options are there

F = 0.194 (sums of squares, not mean squares)
df_error = 11 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 9

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (9)

A randomized complete block design tests 3 asphalt mixes across 5 blocks. The total sum of squares is 470.0, treatments contribute 65 and blocks 55. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 3 treatments, b = 5 blocks$
$SS_total = 470.0$
$SS_treat = 65$
$SS_block = 55$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 470.0 - 65 - 55 = 350.0$
Degrees of freedom
$df_treat = 2, df_block = 4, df_error = 8$
Mean squares
$MS_treat = 65/2 = 32.50; MS_error = 350.0/8 = 43.75$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 32.50/43.75 = 0.743$

Answer:

SS_error = 350.0, F = 0.74 with (2, 8) degrees of freedom

Why the other options are there

F = 0.186 (sums of squares, not mean squares)
df_error = 14 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design

Example 10

Randomized complete block design: ANOVA table and F statistic — Randomized Complete Block Design (10)

A randomized complete block design tests 3 asphalt mixes across 4 blocks. The total sum of squares is 320.0, treatments contribute 175.0 and blocks 110.0. Build the ANOVA table and compute the treatment F statistic.

Given

$t = 3 treatments, b = 4 blocks$
$SS_total = 320.0$
$SS_treat = 175.0$
$SS_block = 110.0$

Find

Error sum of squares, mean squares and F

Start with the thinking

In a randomized complete block design the error term is what remains after treatments and blocks.
Degrees of freedom for error are (t − 1)(b − 1).

Step-by-step solution

Formula
$SS_error = SS_total - SS_treat - SS_block$
Substituting
$SS_error = 320.0 - 175.0 - 110.0 = 35.0$
Degrees of freedom
$df_treat = 2, df_block = 3, df_error = 6$
Mean squares
$MS_treat = 175.0/2 = 87.50; MS_error = 35.0/6 = 5.83$
Formula
$F = MS_treat/MS_error$
Substituting
$F = 87.50/5.83 = 15.000$

Answer:

SS_error = 35.0, F = 15.00 with (2, 6) degrees of freedom

Why the other options are there

F = 5.000 (sums of squares, not mean squares)
df_error = 11 (total df used)

Reference: FE Reference Handbook — Probability and Statistics → Randomized Complete Block Design