1.8 Estimate the mean, variance, and standard deviation for grouped data | Athabasca University MATH 215: Introduction to Statistics

Athabasca University MATH 215: Introduction to Statistics

Descriptive Statistics

Using class midpoints to summarize frequency tables

A grouped frequency table reports how many observations fall in each interval, rather than listing every individual value. For example, it might show how many delivery times were from 10 to less than 20 minutes. This saves space, but it hides the exact values within each interval. To estimate the mean, variance, and standard deviation, we represent every observation in an interval by that interval’s midpoint. The resulting summaries are useful, but they are generally estimates rather than exact calculations from the original data.

What you will learn

Explain why grouped-data summaries use class midpoints and why the results are estimates.
Calculate a grouped mean from class midpoints and frequencies.
Calculate grouped variance and standard deviation, distinguishing sample summaries from population summaries.
Interpret each result with appropriate units and rounding.

1. Read the grouped data and identify what is being estimated

A class is one interval in a grouped table, such as 10 to less than 20 minutes. The frequency, written

f

, is the number of observations in that class. The class midpoint, written

m

, is the value halfway between the two class limits. Find it by adding the limits and dividing by 2. This uses familiar algebra: add two numbers, then divide their sum by 2.

For grouped data, we treat each of the

f

observations in a class as if its value were the midpoint

m

. This is a practical approximation, not a claim that all observations actually equal the midpoint. The total number of observations is the sum of the frequencies, written

n=\sum f

. The symbol

\sum

means “add the listed values.”

The population is the full group of individuals or measurements of interest; a sample is a smaller group collected from that population. A population mean and variance describe the full population. A sample mean and sample variance summarize a sample. The sample mean is often used to estimate the population mean. Whether a table describes a whole population or a sample matters when choosing the variance denominator.

m=\frac{\text{lower class limit}+\text{upper class limit}}{2},\qquad n=\sum f

Use a midpoint for each class and the class frequency as its weight.
Check that frequencies count all observations and that classes do not overlap.
Grouped calculations estimate the summaries of the original ungrouped observations.

2. Estimate the mean

The mean is the arithmetic average: add the values and divide by how many values there are. In a grouped table, each midpoint stands in for every observation in its class. Therefore, multiply each midpoint by its frequency, add those products, and divide by the total frequency. The product

fm

represents the estimated contribution of that class to the total of all observations.

This method is appropriate when the table gives class intervals and frequencies, and the midpoint is a reasonable representative value for each class. The intervals should cover the observations being summarized without gaps or overlaps. If a class is unusually wide, or observations are concentrated near one end of a class, the midpoint approximation may be less representative. The grouped table alone does not reveal how values are spread inside each class.

\bar{x}=\frac{\sum fm}{n}

The grouped mean is a frequency-weighted average of the class midpoints.
The mean has the same units as the measured variable, such as minutes or kilograms.

3. Estimate variance and standard deviation

Variance describes how spread out values are around their mean. To estimate it, find each midpoint’s difference from the grouped mean, square that difference, and multiply by the frequency. Squaring makes negative and positive differences contribute positively and gives more weight to larger differences. Adding these frequency-weighted squared differences gives the estimated total squared spread.

If the grouped table describes the entire population, divide that total by

N

, the population count. The result is the population variance, written

\sigma^2

. If the table describes a sample, the usual sample variance divides by

n-1

and is written

s^2

. The subtraction of 1 is part of the standard sample-variance calculation; do not use it for a full population. In either case, variance is in squared units, such as minutes squared.

The standard deviation is the square root of the variance. It returns the spread to the original measurement units, which makes it easier to interpret alongside the mean. For grouped data, both variance and standard deviation are estimates because they use class midpoints in place of the individual values. Keep unrounded values through the calculation, then round the reported results at the end.

\sigma^2=\frac{\sum f(m-\bar{x})^2}{N},\qquad s^2=\frac{\sum f(m-\bar{x})^2}{n-1},\qquad s=\sqrt{s^2}

Use $N$ for a full population and $n-1$ for a sample variance.
Standard deviation is the positive square root of variance.
Variance uses squared units; standard deviation uses the variable’s original units.

4. A calculation aid and a sensible check

A calculation table helps keep the frequency, midpoint, and squared-deviation work organized. Add the frequency column to find the total count. Add the

fm

column to find the estimated total. After calculating the mean, complete the squared-deviation column using that mean. Keep several decimal places during intermediate arithmetic so rounding does not noticeably alter the final summaries.

A quick reasonableness check can catch common arithmetic errors. The mean should lie between the smallest and largest class midpoints. Variance cannot be negative, and standard deviation cannot be negative. The standard deviation should be zero only if every represented midpoint is the same. Remember that these checks assess the calculation; they do not remove the uncertainty caused by grouping.

Organize calculations by class so each frequency is applied consistently.
Check the count, the mean’s location, and the nonnegative spread measures.

Organizing the delivery-time calculation

Time interval (minutes)	Frequency $f$	Midpoint $m$	$fm$	$f(m-24)^2$
10 to less than 20	3	15	45	243
20 to less than 30	5	25	125	5
30 to less than 40	2	35	70	242
Total	10	—	240	490

Worked example

Estimate delivery-time summaries

A sample of delivery times is grouped as follows. Estimate its mean, sample variance, and sample standard deviation. Also state the corresponding population-style variance and standard deviation if the same table represented the entire population. Report final answers to two decimal places where needed.

Find the midpoints and count
The intervals are nonoverlapping and cover the observations shown. Represent each time in an interval by its midpoint. Add the frequencies to find the sample size.
$m=(15,25,35),\qquad n=3+5+2=10$
Calculate the grouped mean
Multiply each midpoint by its frequency, add those products, and divide by the total count. This produces the estimated average delivery time.
$\bar{x}=\frac{3(15)+5(25)+2(35)}{10}=\frac{240}{10}=24\text{ minutes}$
Find the estimated squared spread
Subtract the mean from each midpoint, square each difference, and multiply by its frequency. The three contributions are $3(15-24)^2=243$ , $5(25-24)^2=5$ , and $2(35-24)^2=242$ . Their sum is $490$ minutes squared.
$\sum f(m-\bar{x})^2=243+5+242=490\text{ minutes}^2$
Calculate sample variance and standard deviation
Because the table describes a sample, divide the estimated squared spread by $n-1=9$ . Take the positive square root of the variance to return to minutes. Keep the unrounded values until reporting the results.
$s^2=\frac{490}{9}=54.4444\ldots\text{ minutes}^2,\qquad s=\sqrt{54.4444\ldots}=7.3786\ldots\text{ minutes}$
Calculate the population-style values
If the same ten grouped observations were the whole population, divide the squared spread by $N=10$ instead. Its population standard deviation is the square root of that population variance.
$\sigma^2=\frac{490}{10}=49\text{ minutes}^2,\qquad \sigma=\sqrt{49}=7\text{ minutes}$

Answer: For the sample, the estimated mean is 24.00 minutes, the sample variance is 54.44 minutes squared, and the sample standard deviation is 7.38 minutes. If the table represents the whole population, the population variance is 49 minutes squared and the population standard deviation is 7 minutes.

Check: The mean of 24 minutes lies between the smallest and largest midpoints, 15 and 35 minutes. Both variance estimates are nonnegative, and the standard deviations are in minutes. The sample variance is larger here because it divides the same squared spread by 9 rather than 10.

Common mistakes and how to avoid them

Using the class limits as if every observation in a class equalled its lower or upper endpoint.

Correction: Use the midpoint as the representative value for each class, and remember this creates an estimate.

Dividing the sample’s squared spread by

n

when calculating the usual sample variance.

Correction: For a sample, divide by

n-1

. For a full population, divide by its count

N

.

Reporting variance in the original measurement units.

Correction: Variance is in squared units. Standard deviation, after taking the square root, is in the original units.

Rounding the mean or squared deviations early and using those rounded values in later steps.

Correction: Keep four to six decimal places in intermediate work when needed, and round only the final reported result.

Lesson summary

Find each class midpoint and use its frequency to represent the observations in that class.
Estimate the mean with $\bar{x}=\sum fm/n$ .
Estimate variance by summing frequency-weighted squared differences from the grouped mean.
Divide by $n-1$ for sample variance or by $N$ for population variance; take the square root for standard deviation.
Interpret grouped summaries as estimates because the table does not show the individual values within each interval.

Check your understanding

Question 1

A grouped sample has midpoints 4 and 10 with frequencies 2 and 3. What is its estimated mean?

7
7.6
8
14

Show answer and explanation

7.6

The total frequency is

2+3=5

, and the weighted total is

2(4)+3(10)=38

. The estimated mean is

38/5=7.6

.

Question 2

A grouped table describes a sample of 8 observations. Which denominator is used for its usual sample variance?

7
8
9
The number of classes

Show answer and explanation

7

Sample variance uses

n-1

, so for

n=8

the denominator is

8-1=7

.

Question 3

If a grouped population variance is 16 minutes squared, what is its standard deviation?

4 minutes
8 minutes
16 minutes
256 minutes

Show answer and explanation

4 minutes

Standard deviation is the positive square root of variance:

\sqrt{16}=4

. Its unit is minutes, not minutes squared.

Key terms

Class: An interval used to group values in a frequency table.
Frequency: The number of observations in a class.
Class midpoint: The value halfway between a class’s lower and upper limits, used as a representative value.
Mean: The arithmetic average of a set of values.
Variance: A measure of spread based on squared differences from the mean.
Standard deviation: The positive square root of variance, expressed in the original measurement units.
Estimate: A calculated value used to approximate a quantity that is not known exactly.

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1.8 · Estimate the mean, variance, and standard deviation for grouped data

Athabasca University MATH 215: Introduction to Statistics

Using class midpoints to summarize frequency tables

What you will learn

1. Read the grouped data and identify what is being estimated

2. Estimate the mean

3. Estimate variance and standard deviation

4. A calculation aid and a sensible check

Organizing the delivery-time calculation

Estimate delivery-time summaries

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MATH 215

About this lesson