Standard Deviation Calculation for First 7 Natural Numbers

The question asks for the standard deviation of the first 7 natural numbers, which are 1, 2, 3, 4, 5, 6, 7.

Step 1: Calculate the Mean

The mean ($\mu$) is the average of the numbers.

Numbers = {1, 2, 3, 4, 5, 6, 7}

Number of observations (N) = 7

Mean ($\mu$) = $\frac{1+2+3+4+5+6+7}{7} = \frac{28}{7} = 4$

Step 2: Calculate the Variance

Variance ($\sigma^2$) is the average of the squared differences from the Mean.

• Squared difference for 1: $(1 - 4)^2 = (-3)^2 = 9$
• Squared difference for 2: $(2 - 4)^2 = (-2)^2 = 4$
• Squared difference for 3: $(3 - 4)^2 = (-1)^2 = 1$
• Squared difference for 4: $(4 - 4)^2 = (0)^2 = 0$
• Squared difference for 5: $(5 - 4)^2 = (1)^2 = 1$
• Squared difference for 6: $(6 - 4)^2 = (2)^2 = 4$
• Squared difference for 7: $(7 - 4)^2 = (3)^2 = 9$

Sum of squared differences = $9 + 4 + 1 + 0 + 1 + 4 + 9 = 28$

Variance ($\sigma^2$) = $\frac{\text{Sum of squared differences}}{N} = \frac{28}{7} = 4$

Step 3: Calculate the Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance.

Standard Deviation ($\sigma$) = $\sqrt{\text{Variance}} = \sqrt{4} = 2$

Therefore, the standard deviation of the first 7 natural numbers is 2.