Understanding the Normal Distribution
The normal distribution is a fundamental concept in statistics characterized by its bell-shaped curve, defined by its mean and standard deviation.
Given Information
- Mean (μ) = 50
- Value greater than 60 = 5% of the distribution
Calculating the Z-Score
To find the Z-score corresponding to the value of 60, we can use the properties of the normal distribution:
- The Z-score represents how many standard deviations a value is from the mean.
- Since 5% of the values are above 60, this corresponds to the 95th percentile of the distribution.
Finding Z-Score for 95th Percentile
- From Z-tables or standard normal distribution tables, the Z-score that corresponds to the 95th percentile is approximately 1.645.
Using the Z-Score Formula
The Z-score formula is:
- Z = (X - μ) / σ
Where:
- X = value (60 in this case)
- μ = mean (50)
- σ = standard deviation
Setting Up the Equation
By substituting the known values into the Z-score formula, we can solve for σ (the standard deviation):
- 1.645 = (60 - 50) / σ
- 1.645 = 10 / σ
Calculating Standard Deviation
Rearranging the equation gives:
- σ = 10 / 1.645
- σ ≈ 6.08
Coefficient of Variation (C/V)
The Coefficient of Variation (C/V) is calculated as:
- C/V = (σ / μ) × 100
Substituting the values:
- C/V = (6.08 / 50) × 100 ≈ 12.16%
This C/V indicates the relative variability of the distribution concerning its mean.