Question

What are the mean and standard deviation of the standard normal distribution? (b) What would be the mean and standard deviation of a distribution created by multiplying the standard normal distribution by 8 and then adding \(75 ?\)

Short answer

Mean is 75; standard deviation is 8.

Step-by-step solution

Define the standard normal distribution

The standard normal distribution, also known as the Z-distribution, is a special type of normal distribution that has a mean (\( \mu \) ) of 0 and a standard deviation (\( \sigma \) ) of 1.

Understand transformation of normal distributions

When you multiply a normal distribution by a constant and then add a constant, the mean and standard deviation are transformed. Multiplying by a constant scales the standard deviation, while adding a constant shifts the mean.

Calculate for transformed mean

Multiply the mean of the standard normal distribution (which is 0) by 8, then add 75. Thus, the new mean is: \[ \text{New Mean} = 0 \times 8 + 75 = 75 \]

Calculate for transformed standard deviation

Multiply the standard deviation of the standard normal distribution (which is 1) by 8. Thus, the new standard deviation is: \[ \text{New Standard Deviation} = 1 \times 8 = 8 \]

Key concepts

Mean Transformation

Mean transformation involves shifting the entire distribution along the horizontal axis. In a normal distribution, the mean (\(\mu\)) is the center point or average of all data points. If you add a constant to every single data point in the distribution, the mean of the distribution shifts to a new value.
This happens because the mean represents the balance point of the data, and adding a constant shift moves this balance point by the same constant amount.

• For the standard normal distribution, the original mean is 0.
• If each data value is increased by adding 75, the new mean becomes 75.

This transformation does not alter the shape of the distribution in any way, only its position along the axis.

Standard Deviation Transformation

The standard deviation (\(\sigma\)) of a distribution tells us how spread out the data points are from the mean. A transformation of the standard deviation involves changing this spread. When multiplying each data point by a constant, the standard deviation is scaled by that constant.
This transformation impacts how tightly or widely data points cluster around the mean.

• For a standard normal distribution, the initial standard deviation is 1.
• If the distribution is multiplied by 8, each data point becomes 8 times further from the mean.

The result is a new standard deviation of 8, indicating a wider spread of data points.
Standard deviation transformation affects the distribution's width but not its central location on the axis.

Normal Distribution

The normal distribution is a fundamental concept in statistics and probability, often called the "bell curve" due to its shape. This distribution is characterized by its symmetry around the mean, a specific mean (\(\mu\)) and standard deviation (\(\sigma\)), and the characteristic bell-shaped curve.
Key attributes of a normal distribution include:

• Symmetrical shape, meaning half the data lies below and half above the mean.
• The mean, median, and mode of a normal distribution are all equal.
• Approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three (empirical rule).

The standard normal distribution is a specific type of normal distribution with a mean of 0 and a standard deviation of 1. This makes it extremely useful for standardizing any normal distribution, making comparisons possible across different data sets with varying means and variances. Transformations of these distributions allow statisticians to adjust the data without changing the underlying probability characteristics, which is critical for many statistical procedures and analyses.