Question

A set of test scores are normally distributed. Their mean is 100 and standard deviation is \(20 .\) These scores are converted to standard normal z scores. What would be the mean and median of this distribution? a. 0 b. 1 c. 50 d. 100

Short answer

The mean and median are both 0; so the answer is a.

Step-by-step solution

Understanding the Problem

We are given a set of test scores that are normally distributed with a mean of 100 and a standard deviation of 20. We need to find the mean and median of the standard normal distribution of these scores.

Recalling Characteristics of Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. When data is converted into z-scores using their mean and standard deviation, they become part of the standard normal distribution.

Calculating Z-scores

The z-score for a data point is calculated using the formula \( z = \frac{x - \mu}{\sigma} \), where \( x \) is a score from the data set, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. This transformation standardizes the data to have a mean of 0 and a standard deviation of 1.

Determining Mean and Median of Z-scores

After converting scores to z-scores, the distribution will reflect characteristics of the standard normal distribution. Specifically, the mean and median of the standardized distribution will both be 0.

Verifying with Options

Based on the standard normal distribution properties, the correct option is the one that asserts the mean and median are both 0. Therefore, the correct answer to the multiple-choice question is option a which states 0.

Key concepts

Understanding the Standard Normal Distribution

The standard normal distribution is a special type of normal distribution. This distribution has a very specific mean of 0 and a standard deviation of 1. This means that when data is centered around these values, it allows us to easily interpret how values compare to one another. To convert any set of normally distributed data into this standard format, we use a process called 'standardization'. Each data point in the set is transformed into what is known as a z-score. This mathematical transformation aligns the mean of the data with 0 and scales its spread to have a standard deviation of 1. Converting data to z-scores is a way of normalizing it, which means it places it in a standard format that is universally comparable. This is particularly useful when comparing data from different sets or distributions.

An Overview of the Normal Distribution

The concept of the normal distribution is a fundamental element in statistics. It is often referred to as a bell-shaped curve due to its symmetric shape centered around the mean. A normal distribution is defined by two parameters:

• The mean, which is the average of all the data points, and
• The standard deviation, which indicates how spread out the data points are around the mean.

Most data points in a normal distribution tend to cluster around the mean. As you move away from the mean, the frequency of the data points decreases. This pattern is what gives the distribution its 'bell shape'. Understanding the normal distribution is essential because many naturally occurring phenomena follow this pattern, making it a crucial concept in both fields of statistics and probability. When data follows a normal distribution, we can make meaningful inferences about the entire set using techniques like calculating probabilities and z-scores.

Diving Into Mean and Median

When dealing with normally distributed data, the mean and median are two vital statistics that summarize the data set. - **Mean**: This is essentially the average. To find it, sum all data points and divide by the number of points. In a perfectly normal distribution, the mean locates at the center of the bell curve. - **Median**: This is the middle value. When you organize all data points in order, the median is the data point that splits the set into two equal halves. In normal distributions, both the mean and the median are equal. They sit at the center of the distribution. This is an important property, as it indicates the symmetry of the data spread. Hence, when test scores are converted to z-scores in our example, both the mean and median of the standardized scores become 0, perfectly aligning with the properties of the standard normal distribution.