If N=10, ΣX=80, ΣX²=1000, then standard deviation will be :

Question

Accepted Answer

The correct option is 1 (6). The standard deviation (σ) is a measure of the amount of variation or dispersion of a set of values. It is calculated using the formula: σ = SQRT((ΣX² / N) - (ΣX / N)²). First, we calculate the arithmetic mean (X̄), which is ΣX / N, or 80 / 10 = 8. Next, we substitute the values into the formula: σ = SQRT((1000 / 10) - (80 / 10)²). This simplifies to σ = SQRT(100 - 8²), which is σ = SQRT(100 - 64). Therefore, σ = SQRT(36), which equals 6. The standard deviation measures how spread out the numbers are from the mean; in this case, the spread is 6. The incorrect option is 2 (8). This value represents the arithmetic mean (X̄ = 80/10 = 8) of the data set, not the measure of dispersion, which is the standard deviation. The incorrect option is 3 (10). This value is derived from calculating the square root of the first part of the variance formula (SQRT(ΣX²/N) = SQRT(1000/10) = 10), but it ignores the mean squared component (X̄²), which is essential for calculating the correct standard deviation. The incorrect option is 4 (12). This value does not correspond to any known statistical measure derived from the given inputs and is significantly higher than the actual standard deviation of 6. A deviation of 12 would suggest a much wider spread in the data than the given sums imply.