Variance – SS3 Further Mathematics Lesson Note (First Term)

The assumed mean method is a short way of finding the mean, variance and standard deviation when the data contain large numbers.

Behavioural objectives

By the end of the lesson, students should be able to:

1. explain why an assumed mean is used for large numbers;
2. calculate the mean of grouped data using an assumed mean;
3. calculate the variance of grouped data using an assumed mean;
4. calculate the standard deviation of grouped data;
5. state how a coded factor simplifies the working.

What are variance and standard deviation?

Variance is the mean of the squared deviations of the items from their mean.

• Variance: measures how widely the data are spread about the mean. It is written σ² or s².
• Standard deviation: the positive square root of the variance. It is written σ or s.
• Small value: the items are close to the mean.
• Large value: the items are widely scattered.

For grouped data, the direct formula is: variance = Σf(x − x̄)² / Σf.

What is an assumed mean?

An assumed mean A is a convenient value chosen near the middle of the data to make the deviations small.

• A (assumed mean): a guess near the centre, usually the class mark of a middle class with a large frequency.
• x: the value of the item, or the class mark (midpoint) of a class.
• f: the frequency of each item or class.
• d: the deviation of each item from the assumed mean, d = x − A.
• d̄ (mean of the deviations): Σfd / Σf.

Any value of A gives the correct answer. A value near the true mean gives smaller numbers.

Why is an assumed mean used?

An assumed mean is used because small deviations are easier to square and add than large raw values.

• Smaller numbers: d is much smaller than x, so there is less multiplication.
• Fewer errors: the arithmetic is shorter and easier to check.
• No need for the exact mean first: the mean is found at the end by adding d̄ to A.

What are the formulas for the assumed mean method?

The mean is x̄ = A + d̄, and the variance is Σfd²/Σf minus the square of d̄.

Mean

• Mean of deviations: d̄ = Σfd / Σf.
• Actual mean: x̄ = A + d̄ = A + Σfd / Σf.

Deviation from the mean

• Result: x − x̄ = d − d̄, because the same number A is taken away from both x and x̄.
• Meaning: subtracting A does not change the spread of the data.
• Effect: the variance of d equals the variance of x.

Variance and standard deviation

• Variance: σ² = Σfd²/Σf − (Σfd/Σf)².
• Standard deviation: σ = √[Σfd²/Σf − (Σfd/Σf)²].
• Note: Σfd² means square d first, then multiply by f. It is not (Σfd)².

Using a coded factor

A coded factor C is a common factor of all the deviations, used to divide them and make them smaller still.

• Coded value: u = (x − A) / C. This is useful when the class width is the same for all classes.
• Mean: x̄ = A + C × ū, where ū = Σfu / Σf.
• Variance: σ² = C² × [Σfu²/Σf − (Σfu/Σf)²].
• Standard deviation: σ = C × √[Σfu²/Σf − (Σfu/Σf)²].

What are the steps for calculating the variance using an assumed mean?

1. Find the class mark x of each class.
2. Choose an assumed mean A near the middle of the data.
3. Find d = x − A for each class.
4. Multiply each d by its frequency to get fd.
5. Find d² and multiply by f to get fd².
6. Add the columns to get Σf, Σfd and Σfd².
7. Find the mean: x̄ = A + Σfd/Σf.
8. Find the variance: Σfd²/Σf − (Σfd/Σf)².
9. Take the square root of the variance to get the standard deviation.

Worked example: how do we find the variance and standard deviation of shoe sizes?

The data below show the shoe sizes of 100 men for the NYSC.

Calculate for the data the:

1. variance,
2. standard deviation.

Solution

Take the assumed mean A = 62, the class mark of the class with the highest frequency.

1. Mean of deviations: d̄ = 190 / 100 = 1.9.
2. Mean: x̄ = 62 + 1.9 = 63.9 mm.
3. Mean of squared deviations: Σfd²/Σf = 5000 / 100 = 50.
4. Variance: σ² = 50 − (1.9)² = 50 − 3.61 = 46.39.
5. Standard deviation: σ = √46.39 = 6.81 mm (2 decimal places).

What mistakes should students avoid?

• Forgetting to subtract d̄²: Σfd²/Σf alone is not the variance, unless A equals the mean.
• Using class limits: always use the class mark x in the calculation.
• Losing negative signs: d is negative for classes below A. d² is always positive.
• Leaving the mean as d̄: add A back to get the mean x̄.
• Forgetting the square root: the standard deviation is the root of the variance.
• Forgetting C: when coding, multiply the variance by C², and the standard deviation by C.

Evaluation

The table below shows the transport allowance paid to 50 senior staff of a computer manufacturing company.

Calculate the following for the data:

1. the arithmetic mean, using an assumed mean of ₦1025,
2. the variance,
3. the standard deviation.

Assignment

The weights in kg of 200 newborn babies in a local government area are given in the table below.

Using an assumed mean of 55 kg, calculate the mean, the variance and the standard deviation for the data.