Step-Deviation vs Assumed Mean Method | Class 10 Statistics

Quick answer: The assumed mean method uses $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$ with $d_i = x_i - a$. The step-deviation method also divides each deviation by the class size $h$: $u_i = \frac{x_i - a}{h}$ and $\bar{x} = a + \left(\frac{\sum f_i u_i}{\sum f_i}\right)h$. Both always give the same mean; step-deviation just keeps the numbers smaller. It is convenient only when all the deviations share a common factor $h$.

Three ways to find the mean of grouped data

Here $x_i$ is the class mark (mid-point) of each class, $a$ is the assumed mean (usually a class mark near the middle), and $h$ is the class size.

When is the step-deviation method not convenient?

To use step-deviation, every deviation $d_i$ must be divided by the same number $h$.

• If all classes have the same width, the deviations are all multiples of that width, so the method works neatly.
• If the class widths are unequal and the deviations have no common factor, dividing by $h$ gives awkward fractions and saves no work. Use the assumed mean or direct method instead.

Key point: All three methods give exactly the same mean. Assumed mean and step-deviation are only shortcuts that make the arithmetic easier.

Solved example (NCERT): find the missing frequency

The table shows the daily pocket allowance of children in a locality. The mean allowance is Rs 18. Find the missing frequency $f$.

Step 1: Class marks, assumed mean and class size.

Class marks: 12, 14, 16, 18, 20, 22, 24. Class size $h = 2$. Take the assumed mean $a = 18$.

Step 2: Make the table.

Step 3: Assumed mean method.

$18 = 18 + \frac{2f - 40}{44 + f}$

So:

$\frac{2f - 40}{44 + f} = 0$

$2f - 40 = 0 \quad \Rightarrow \quad f = 20$

Step 4: Check with the step-deviation method.

$18 = 18 + \left(\frac{f - 20}{44 + f}\right) \times 2$

This again gives $f - 20 = 0$, so $f = 20$. ✓

Step 5: Verify the mean.

With $f = 20$: $\sum f_i = 64$, $\sum f_i d_i = 40 - 40 = 0$ and $\sum f_i u_i = 0$.

$\bar{x} = 18 + \frac{0}{64} = 18$

Answer: The missing frequency is 20. Both methods give the same result, and the mean comes out to Rs 18 as stated.

Summary table

Exam tips

• Choose $a$ as a class mark near the middle of the table; it makes the deviations small and symmetric.
• In the step-deviation formula, multiply by $h$ before adding $a$.
• Add the negative products and the positive products separately, then combine them. This avoids sign errors.
• If the question names a method, you must use that method even if another is quicker.

Common mistakes

• Forgetting to multiply by $h$ at the end of the step-deviation method.
• Using the class limits instead of the class marks for $x_i$.
• Choosing a value for $a$ that is not a class mark, which makes the deviations awkward.
• Thinking the two methods can give different means. If they do, there is an arithmetic error.

Frequently Asked Questions

What is the difference between the assumed mean and step-deviation methods?

The assumed mean method uses deviations dᵢ = xᵢ − a. The step-deviation method divides these deviations by the class size h, giving uᵢ = (xᵢ − a)/h, and multiplies by h at the end.

Do both methods give the same mean?

Yes. All three methods, direct, assumed mean and step-deviation, always give the same mean. The shortcuts only reduce the size of the numbers.

When should the step-deviation method not be used?

It is not convenient when the deviations do not share a common factor, which can happen when class widths are unequal. Use the assumed mean or direct method then.

How do you choose the assumed mean?

Pick a class mark near the middle of the distribution, ideally one with a large frequency. This keeps the deviations small.

What is the missing frequency in the NCERT pocket allowance question?

The missing frequency is 20. With f = 20, the sum of fᵢdᵢ is zero, so the mean equals the assumed mean, Rs 18.