The standard normal distribution is a special type of normal distribution where the mean is 0, and the standard deviation is 1. It simplifies probability calculations and comparisons across datasets by converting raw data into Z-scores. Below, we explore its properties, calculations, and practical examples.
This graph illustrates the standard normal distribution with probabilities (p-values) between standard deviations.
Bob is 200 cm tall, while the mean height of people in Germany is 170 cm, with a standard deviation of 10 cm. To calculate Bob’s Z-value:
Z = (X – μ) / σ = (200 – 170) / 10 = 3
Bob’s height is 3 standard deviations above the mean. Using a Z-table or software, we find that only 0.13% of Germans are taller than Bob.
To find the proportion of people in Germany between 155 cm and 165 cm:
Using Z-tables or software, the probabilities are:
Subtracting these values gives: 30.85% – 6.68% = 24.17%
To find how tall you must be to be taller than 90% of Germans:
Z = 1.281
Using the Z formula:
X = Z * σ + μ = 1.281 * 10 + 170 = 182.81 cm
You need to be at least 182.81 cm tall to be taller than 90% of Germans.
Explore the properties of the standard normal distribution and its applications in real-world problems.