The step deviation method

What you will be able to do

• find the step deviation of each class;
• find the mean by the step deviation method;
• choose between the direct, assumed mean and step deviation methods.

Smaller still

Sixty delivery vans drove these distances in one day. The classes are $20$ km wide.

Take the assumed mean $a = 110$. The deviations are $-40, -20, 0, 20, 40, 60$. They are all multiples of $20$, which is the class width $h$. This always happens when the classes have equal width and $a$ is a class mark, because neighbouring class marks are exactly $h$ apart. So divide each deviation by $h$. The result is the step deviation:

The step deviations here are $-2, -1, 0, 1, 2, 3$. They count how many classes a class is away from the class of $a$.

The formula

Write $x = a + hu$ and add up as before. Then $\sum fx = aN + h\sum fu$, and

The class width $h$ comes back at the end, because we divided by $h$ at the start.

Check with the direct method: $\sum fx = 350 + 810 + 1980 + 1820 + 1350 + 850 = 7160$, and $\dfrac{7160}{60} = 119.33$ km. The two agree.

Which method should I use?

All three methods give the same mean. They differ in how much work each needs.

A warning. The step deviation method needs equal class widths. If the widths differ, the numbers $\dfrac{x - a}{h}$ are not whole numbers, and the method loses its advantage. Use the assumed mean method or the direct method then.

Another warning: keep the factor $h$. After finding $\dfrac{\sum fu}{N}$ you must multiply by $h$ and add $a$.

Practise with new numbers

Try it in Python

This program finds the mean of the van table by all three methods and compares the answers. It also asks NumPy and the statistics module, as a fourth and a fifth check.

Common mistakes

• Forgetting the class width at the end. The mean is $a + h\dfrac{\sum fu}{N}$, not $a + \dfrac{\sum fu}{N}$.
• Using the method with unequal classes. The step deviation method needs the same width for all classes.
• Dividing by $h$ in the wrong place. Divide each deviation by $h$ before multiplying by the frequency, not after the sum.
• Mixing up $d$ and $u$. The deviation $d = x - a$ has the unit of the data. The step deviation $u = \dfrac{d}{h}$ has no unit.

Remember these

• $u = \dfrac{x - a}{h}$, and the step deviations are small whole numbers when the classes have equal width.
• $\bar{x} = a + h\dfrac{\sum fu}{\sum f}$.
• Direct, assumed mean and step deviation always give the same mean. Choose by the size of the numbers.
• The step deviation method needs equal class widths.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The class marks are $30, 40, 50, 60, 70$ with $a = 50$ and $h = 10$. Find the step deviations.

2. For $N = 60$, $a = 45$, $h = 5$ and $\sum fu = -12$, find the mean.

3. Why does the step deviation method need all classes to have the same width?

4. Class marks such as $1005, 1015, 1025, \dots$ are very large. Which method would you choose?

5. With $a = 25$, $h = 10$, $N = 30$ and $\sum fu = 18$, find the mean.

References

• NCERT, Mathematics for Class 10, the chapter Statistics: the step deviation method.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics. A-Level texts call the same idea coding.