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Add 5 marks to every score, convert temperatures from Celsius to Fahrenheit, or measure in different units, and every statistic seems to need recalculating. It does not. Two rules tell you what happens to the mean, median, range, standard deviation and variance, and NCERT's step-deviation shortcut is built on the same rules. This post proves both rules, lists which measures move and which do not, and works the shortcut on grouped data.
Where is this in the syllabus?
The NCERT Class 11 chapter Statistics (reprint 2026-27) has a section called "Shortcut method to find variance and standard deviation". It sets yᵢ = (xᵢ − A)/h and proves that x̄ = A + hȳ and σx² = h²σy². Its Miscellaneous Examples show that multiplying every value by a constant multiplies the variance by the constant squared, and that adding a constant leaves it unchanged. The NTA Syllabus for JEE (Main) 2026 lists variance and standard deviation for grouped and ungrouped data in Unit 13. Check the syllabus for your exam year.
Why does adding a constant not change the spread?
Let y = x + b. Then ȳ = x̄ + b, and every deviation y − ȳ = (x + b) − (x̄ + b) = x − x̄ is unchanged. So the variance, standard deviation, mean deviation and range stay the same. Every measure of centre (mean, median, mode) moves by b.
Why does scaling multiply the variance by a²?
Let y = ax. Then ȳ = a·x̄, each deviation becomes a(x − x̄), and each squared deviation is multiplied by a². So the variance is multiplied by a², and the standard deviation by |a|. Combining both, for y = ax + b:
Mean, median and mode: multiplied by a, then b added.
Range, mean deviation and standard deviation: multiplied by |a|.
Variance: multiplied by a².
Worked example 1: step deviation on grouped data
Sample data: heights of 40 students, with class marks 140, 145, 150, 155 and 160 cm and frequencies 2, 8, 15, 10 and 5. Take A = 150 and h = 5, so u = (x − 150)/5 takes the values −2, −1, 0, 1, 2.
fu: −4, −8, 0, 10, 10. Σfu = 8.
fu²: 8, 8, 0, 10, 20. Σfu² = 46.
ū = 8/40 = 0.2, and σu² = 46/40 − 0.2² = 1.15 − 0.04 = 1.11.
Undo the change: x̄ = 150 + 5 × 0.2 = 151 cm, σx² = 25 × 1.11 = 27.75 cm², and σx ≈ 5.27 cm.
Working with the raw class marks gives the same three numbers. Note what A does: it is added back to the mean only. The variance never sees it.
Worked example 2: a change of units
A week of noon temperatures (sample numbers) has mean 30 °C and standard deviation 2 °C. NIST gives the conversion as t/°C = (t/°F − 32)/1.8, so t/°F = 1.8 t/°C + 32.
Mean: 1.8 × 30 + 32 = 86 °F.
Standard deviation: 1.8 × 2 = 3.6 °F.
Variance: 1.8² × 4 = 12.96 (°F)².
The 32 shifts the mean and never appears in the spread.
Worked example 3: working backwards
The values y = 3x − 5 have mean 16 and variance 18. Find the mean and variance of x. From 16 = 3x̄ − 5, x̄ = 7. From 18 = 3²σx², σx² = 2.
A negative factor. If y = 100 − 2x and x has mean 151 and standard deviation 5.27 (from example 1), then ȳ = 100 − 302 = −202, σy = 2 × 5.27 ≈ 10.54 (not −10.54), and the variance is 4 × 27.75 = 111.
Which mistakes cost marks?
Adding A to the variance when undoing the step deviation. A belongs to the mean only.
Multiplying the variance by h instead of h².
A negative standard deviation after scaling by a negative number. Use |a|.
Forgetting the shift in the mean, such as the 32 in a Fahrenheit conversion.
Thinking every measure scales the same way. Variance is the one that gets a².
For the basic formulas and the "wrongly copied value" type, see our posts on mean, variance and standard deviation and correcting a mean and variance.
A practice routine
Do NCERT's Example 12 twice, by the direct method and by step deviation. The answers must agree.
Redo worked example 1 with A = 155. You should get Σfu = −32, ū = −0.8, σu² = 1.11 and the same mean of 151 cm. The variance does not depend on the choice of A.
Write three conversions of your own (marks out of 40 rescaled to 100, rupees to paise, °C to °F) and predict the new mean and standard deviation before calculating.
Do two "work backwards" questions like example 3 each week.
When a step in the table does not come out, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Today, take worked example 1, multiply every height by 2 and add 10, and predict the new mean and variance before you check them.
