Probability & Statistics 1

Mathematics9709

Probability & Statistics 1

Mathematics9709

Variance from totals and combined data

A standard-deviation calculation needs only three totals: how many values there are, their sum, and their sum of squares. That lets you combine data sets or run a question backwards.

Pool totals, not standard deviations

pool these three

n · Σx · Σx²

Add the matching total from each group.

calculate once

variance = Σx²/n − (Σx/n)²

finish

SD = √variance

Never average or add two SDs.

For a combined data set, add the three totals first. Then run the variance formula once. Variances and standard deviations themselves do not add.

Recover a missing squared total

The three totals are the number of values, their sum and their sum of squares. If the mean and standard deviation are given, square the SD to obtain the variance and rearrange the same formula.

\sum x=n\bar x

\sum x^2=n\big(\operatorname{SD}^2+\bar x^2\big)

Worked example

Find the missing group's squared total

A combined set has $n=50$ , mean 8.54 and SD 1.38. One group contributes $\sum x^2=1823$ . Find the other group's squared total.

The whole set has squared total $50(1.38^2+8.54^2)=3741.8$ .

Answer

\sum y^2=3741.8-1823=1918.8

(9709/51/M/J/25 Q3(d))

Combine by adding totals

Create one total count. Add the two group sizes.
Create one ordinary total. Add the two $\sum x$ values. If a mean is given, first recover its total with $n\bar x$ .
Create one squared total. Add the two $\sum x^2$ values. If an SD is given, first recover its squared total.
Run one variance formula. Use the combined totals and the exact combined mean.

Worked example

Standard deviation from pooled totals

A combined set has $n=54$ , $\sum x=1774.8$ and $\sum x^2=58431.72$ .

\bar x=\frac{1774.8}{54}=32.8666\ldots

Keep that unrounded value inside the variance calculation.

Evaluate the squared-total term:

\frac{\sum x^2}{n}=\frac{58431.72}{54}

Then subtract the square of the unrounded mean: $\frac{58431.72}{54}$ minus $(32.8666\ldots)^2$ .

\operatorname{Var}=1.8511\ldots

Answer

\operatorname{SD}=1.36\ \text{(3 s.f.)}

(9709/51/O/N/25 Q3(b))

Common mistake

Never average two variances or two standard deviations. If the group means differ, the gap between those means adds extra spread to the combined set.

A four-line audit prevents most errors

Line	Write	Check
1	N = n₁ + n₂	all observations counted
2	ΣX = Σx + Σy	ordinary totals, not means
3	ΣX² = Σx² + Σy²	squared totals, not squared sums
4	Var = ΣX²/N − (ΣX/N)²	root only if SD is asked

Your turnIndependent check [5 marks]

Group A has 12 values with mean 6 and SD 2. Group B has 8 values with mean 11 and SD 3. Find the population SD of all 20 values.

Show worked answer

A: $\sum x=72$ , $\sum x^2=12(2^2+6^2)=480$ . B: $\sum y=88$ , $\sum y^2=8(3^2+11^2)=1040$ .

The combined mean is $160/20=8$ .

Answer

1520/20-8^2=12

\sqrt{12}=3.46\ \text{(3 s.f.)}

PreviousStandard deviation NextCoded data: shift and scale

Variance from totals and combined data

A standard-deviation calculation needs only three totals: how many values there are, their sum, and their sum of squares. That lets you combine data sets or run a question backwards.

Pool totals, not standard deviations

pool these three

n · Σx · Σx²

Add the matching total from each group.

calculate once

variance = Σx²/n − (Σx/n)²

finish

SD = √variance

Never average or add two SDs.

For a combined data set, add the three totals first. Then run the variance formula once. Variances and standard deviations themselves do not add.

Recover a missing squared total

The three totals are the number of values, their sum and their sum of squares. If the mean and standard deviation are given, square the SD to obtain the variance and rearrange the same formula.

\sum x=n\bar x

\sum x^2=n\big(\operatorname{SD}^2+\bar x^2\big)

Worked example

Find the missing group's squared total

A combined set has $n=50$ , mean 8.54 and SD 1.38. One group contributes $\sum x^2=1823$ . Find the other group's squared total.

The whole set has squared total $50(1.38^2+8.54^2)=3741.8$ .

Answer

\sum y^2=3741.8-1823=1918.8

(9709/51/M/J/25 Q3(d))

Combine by adding totals

Create one total count. Add the two group sizes.
Create one ordinary total. Add the two $\sum x$ values. If a mean is given, first recover its total with $n\bar x$ .
Create one squared total. Add the two $\sum x^2$ values. If an SD is given, first recover its squared total.
Run one variance formula. Use the combined totals and the exact combined mean.

Worked example

Standard deviation from pooled totals

A combined set has $n=54$ , $\sum x=1774.8$ and $\sum x^2=58431.72$ .

\bar x=\frac{1774.8}{54}=32.8666\ldots

Keep that unrounded value inside the variance calculation.

Evaluate the squared-total term:

\frac{\sum x^2}{n}=\frac{58431.72}{54}

Then subtract the square of the unrounded mean: $\frac{58431.72}{54}$ minus $(32.8666\ldots)^2$ .

\operatorname{Var}=1.8511\ldots

Answer

\operatorname{SD}=1.36\ \text{(3 s.f.)}

(9709/51/O/N/25 Q3(b))

Common mistake

Never average two variances or two standard deviations. If the group means differ, the gap between those means adds extra spread to the combined set.

A four-line audit prevents most errors

Line	Write	Check
1	N = n₁ + n₂	all observations counted
2	ΣX = Σx + Σy	ordinary totals, not means
3	ΣX² = Σx² + Σy²	squared totals, not squared sums
4	Var = ΣX²/N − (ΣX/N)²	root only if SD is asked

Your turnIndependent check [5 marks]

Group A has 12 values with mean 6 and SD 2. Group B has 8 values with mean 11 and SD 3. Find the population SD of all 20 values.

Show worked answer

A: $\sum x=72$ , $\sum x^2=12(2^2+6^2)=480$ . B: $\sum y=88$ , $\sum y^2=8(3^2+11^2)=1040$ .

The combined mean is $160/20=8$ .

Answer

1520/20-8^2=12

\sqrt{12}=3.46\ \text{(3 s.f.)}

PreviousStandard deviation NextCoded data: shift and scale