Mathematics, three units

Two classes share an average — what else do you need to compare them?

Quick answer

You need the spread of the marks, measured by the standard deviation or range, and the median. The average only shows the center point. A small standard deviation means student scores cluster tightly around the average. A large standard deviation means the class is widely split between high and low achievers.

The topic explained

Why the average is only half the story

When two classes share an average, what else do you need to compare them? You need to know the spread. The mean tells you where the center of the scores sits, but it tells you nothing about how far students drift from that center.

Consider two classes with an average test score of 75. In Class A, every student scored between 72 and 78. In Class B, half the students scored 100 and the other half scored 50. The averages are identical, but the classroom realities are completely different.

What the standard deviation adds

The standard deviation measures spread: how far marks lie from the mean, on average. It is measured in the exact same units as the data, such as test points.

A low standard deviation shows a uniform class where most students understand the material at a similar level. A high standard deviation shows a divided class with both high achievers and struggling students. In 11th grade 3-unit math, exam questions rarely ask you to calculate standard deviation by hand. Instead, they give you two values and ask you to explain in words what the difference means.

Mean versus median when outliers appear

The mean is the sum of all scores divided by the number of students. The median is the physical middle value once scores are arranged from lowest to highest. When scores are evenly balanced, the mean and median sit close together.

When an extreme score appears, the mean gets dragged toward it, but the median resists. For example, in the scores 60, 70, 80, 90, 100, both the mean and the median are 80. If the top mark becomes 200, the median remains 80, but the mean jumps to 100. Checking the median reveals whether an average is being skewed by a few extreme scores.

Finding a missing score using the total

Many exam questions ask you to work backward from an average to find an unknown score. The core rule to remember is: Mean multiplied by Count equals Total.

If 20 students have an average score of 74, their total score is 20 times 74, which equals 1,480 points. If a new student joins and scores 95, add 95 to get a new total of 1,575 points. Divide by 21 to find the new average of 75.

Free printable worksheet

Comparing Distributions, Medians, and Means

This worksheet is for 11th-grade students preparing for 3-unit mathematics exams. Solve each problem using the definitions of mean, median, and standard deviation.

Print the worksheet Regenerate new worksheet

Free, no sign-up. The answer key prints on the second page. Regenerating a new worksheet on the same topic needs a free account.

Personal AI lessons and practice, from preschool to university

Free printable worksheetlessoons.com

Comparing Distributions, Medians, and MeansName:

Class A and Class B both have a mean score of 80. Class A has a standard deviation of 4, while Class B has a standard deviation of 16. Which class is more uniform?
Find the mean and the median of the following test scores: 65, 75, 80, 85, 95.
If the score 95 in the previous set is replaced with 120, what is the new mean and the new median?
A student scored 70, 82, and 76 on three quizzes. What score must they get on the fourth quiz to reach an overall average of 80?
A student needs an average of 85 across four exams. Their first three scores are 60, 65, and 70. Can they reach their goal on a test graded out of 100?
Group 1 has 10 students with an average of 70. Group 2 has 30 students with an average of 90. Can you calculate the combined average by doing (70 + 90) / 2? Why or why not?
True or False: A high standard deviation always means that the class received high marks.

Common questions

Do I need to calculate standard deviation by hand on the 3-unit exam?

No. In 3-unit high school exams, the standard deviation is almost always provided. Your task is to interpret what a higher or lower value means when comparing two groups in words.

What is the most common mistake students make with the median?

The most common mistake is picking the middle number before ordering the list. You must sort the data from smallest to largest first. If the count is even, average the two middle numbers.

When should I report the median instead of the mean?

Report the median when the data contains extreme values or outliers. For example, income distributions use medians because a handful of multi-millionaires would skew the mean upward, making it unrepresentative.

How do I average two different class averages?

You cannot simply add the averages and divide by two unless both classes have the exact same number of students. If class sizes differ, multiply each average by its class size, add the totals, and divide by the combined student count.

Still have a question? Ask Lessi

Any question about this subject. Lessi answers right here, step by step.

Try asking

Asking needs a free account; your question is kept while you sign up. Answers are written by AI and can contain mistakes. Lessi doesn’t give medical or legal advice.