The Standard Normal Distribution: Calculating Areas and Percentiles

The Standard Normal Distribution

Understanding the Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is used extensively in statistics to calculate probabilities and percentiles for normally distributed data.

• Z-score: Represents the number of standard deviations a value is from the mean.

• Area under the curve: Corresponds to the probability or proportion of values within a specified range.

Calculating Areas Under the Standard Normal Curve

Areas under the standard normal curve can be found using statistical calculators or tables. The calculator function normalcdf is commonly used to compute probabilities for intervals.

• normalcdf(lower, upper, mean, std. deviation): Calculates the probability that a z-score falls between the lower and upper bounds.

Examples of Area Calculations

• Between z = 0 and z = 1.47: $P(0 < z < 1.47) = \text{normalcdf}(0, 1.47, 0, 1) \approx 0.4292$

• Between z = -0.56 and z = 0: $P(-0.56 < z < 0) = \text{normalcdf}(-0.56, 0, 0, 1) \approx 0.2123$

• To the right of z = 1.93: $P(z > 1.93) = \text{normalcdf}(1.93, 10^{99}, 0, 1) \approx 0.0268$

• To the left of z = -0.78: $P(z < -0.78) = \text{normalcdf}(-10^{99}, -0.78, 0, 1) \approx 0.2177$

Percentages and Percentiles in the Standard Normal Distribution

Percentages and percentiles are used to describe the proportion of values above, below, or between certain z-scores. The invNorm function finds the z-score corresponding to a given percentile.

• Percentage above z = -2.46: $P(z > -2.46) = 0.9931 = 99.31\%$

• Percentage between z = 1.24 and z = 2.09: $P(1.24 < z < 2.09) = 0.0892 = 8.92\%$

Finding Percentiles and Cutoff Scores

• 40th percentile (P40): $\text{invNorm}(0.40, 0, 1) \approx -0.25$

• Lowest z-score in the top 2.5%: $\text{invNorm}(0.975, 0, 1) \approx 1.96$

• Z-scores cutting off the lowest 15% and highest 20%:

  • Lowest 15%: $\text{invNorm}(0.15, 0, 1) \approx -1.04$

  • Highest 20%: $\text{invNorm}(0.80, 0, 1) \approx 0.84$

Summary Table: Standard Normal Distribution Calculations

Calculation	Formula	Result
Area between z = 0 and z = 1.47	$\text{normalcdf}(0, 1.47, 0, 1)$	0.4292
Area between z = -0.56 and z = 0	$\text{normalcdf}(-0.56, 0, 0, 1)$	0.2123
Area to the right of z = 1.93	$\text{normalcdf}(1.93, 10^{99}, 0, 1)$	0.0268
Area to the left of z = -0.78	$\text{normalcdf}(-10^{99}, -0.78, 0, 1)$	0.2177
Percentage above z = -2.46	$P(z > -2.46)$	99.31%
Percentage between z = 1.24 and z = 2.09	$P(1.24 < z < 2.09)$	8.92%
40th percentile (P40)	$\text{invNorm}(0.40, 0, 1)$	-0.25
Lowest z-score in top 2.5%	$\text{invNorm}(0.975, 0, 1)$	1.96
Lowest 15% cutoff	$\text{invNorm}(0.15, 0, 1)$	-1.04
Highest 20% cutoff	$\text{invNorm}(0.80, 0, 1)$	0.84