Match List - I with List - II.

List – I (Population Mean (μ) and \(\rm \frac{1}{N}\Sigma x_i^2)\)		List - II (Population Standard Deviation (σ))
A.	\(\rm \mu=5, \frac{1}{N}x_i^2=50\)	I.	8
B.	\(\rm \mu=4, \frac{1}{N}x_i^2=52\)	II.	7
C.	\(\rm \mu=3, \frac{1}{N}x_i^2=58\)	III.	6
D.	\(\rm \mu=6, \frac{1}{N}x_i^2=100\)	IV.	5

Choose the correct answer from the options given below : 

The correct answer is - A - IV, B - III, C - II, D - I

Key Points

• Population Standard Deviation Calculation
  • The formula for the population standard deviation (σ) is derived from the population mean (μ) and the mean of the squared deviations ( $\frac{1}{N}\Sigma x_i^2$ ).
  • We use the formula: $\sigma = \sqrt{\frac{1}{N}\Sigma x_i^2 - \mu^2}$ .
  • For each pair, we calculate σ to match the correct option:
    • A: $\sigma = \sqrt{50 - 5^2} = \sqrt{50 - 25} = \sqrt{25} = 5$
    • B: $\sigma = \sqrt{52 - 4^2} = \sqrt{52 - 16} = \sqrt{36} = 6$
    • C: $\sigma = \sqrt{58 - 3^2} = \sqrt{58 - 9} = \sqrt{49} = 7$
    • D: $\sigma = \sqrt{100 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$

Additional Information

• Population Mean ( $\mu$ )
  • The population mean ( $\mu$ ) is the average of all the values in the population.
  • It is calculated as $\mu = \frac{1}{N}\sum_{i=1}^{N} x_i$ .
• Mean of Squared Deviations ( $\frac{1}{N}\Sigma x_i^2$ )
  • This value represents the average of the squares of the individual data points.
  • It is used in the calculation of the variance and standard deviation.
• Variance ( $\sigma^2$ )
  • Variance is the average of the squared differences from the mean.
  • It is calculated as $\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2$ .
  • The standard deviation is the square root of the variance.