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SD of marks; effect of adding one

Short answer

SD = √(Σfx²/N − (Σfx/N)²) = 1.774 marks. Adding 1 to every mark shifts the mean but does not alter deviations; the SD stays 1.774.

Calculate the standard deviation from the following data regarding marks obtained by students in a test:

Marks123456789
No. of Students324157981238346173
What will be the value of standard deviation if the marks obtained by each of the students are increased by one?

Formula for a frequency distribution: σ = √[ Σfx²/N − (Σfx/N)² ].

Marks xffxfx²
1323232
24182164
357171513
4983921568
51236153075
6834982988
7463222254
8171361088
9327243
Total500227511,925
Mean = 2275/500 = 4.55
σ² = 11,925/500 − (4.55)² = 23.85 − 20.7025 = 3.1475
σ  = √3.1475 = 1.774

(With divisor N − 1 the SD would be 1.776.)

Effect of adding one to every mark

Adding a constant to every observation shifts the mean by the same constant but does not change the distances from the mean. So the SD is unchanged: σ(x + 1) = σ(x). The new mean would be 5.55.

Answer: SD = 1.774 marks; after increasing each mark by one, the SD is still 1.774.

✓ Answer checked by Claude Sonnet 5.5

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