Question

True/false: A Z-score represents the number of standard deviations above or below the mean.

Short answer

True.

Step-by-step solution

Understand the Concept of Z-score

A Z-score, also known as a standard score, indicates how many standard deviations an element is from the mean of a dataset. It is calculated using the formula \( Z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Evaluate the Statement

The statement claims that a Z-score represents the number of standard deviations above or below the mean. This aligns with the concept of a Z-score, as it precisely measures the distance from the mean in terms of standard deviations.

Determine the Truth Value

Since the statement is an accurate description of what a Z-score is and how it functions, the statement is true.

Key concepts

Standard Score

A standard score, commonly referred to as a Z-score, provides a way to understand how a particular data point relates to the mean of a dataset. This is especially useful in standardizing data, allowing for comparisons between different datasets with varying units or scales.
A Z-score indicates how many standard deviations a value is away from the mean. For instance, a Z-score of 1 means the value is one standard deviation above the mean, while a Z-score of -1 means it is one standard deviation below. The formula to calculate a Z-score is:

• \( Z = \frac{(X - \mu)}{\sigma} \)

Here, \( X \) is the value in question, \( \mu \) represents the mean of the dataset, and \( \sigma \) is the standard deviation.
The usefulness of Z-scores lies in their ability to establish a common language for discussing variations from the mean. This comes in handy when assessing the likelihood of occurrences in a normal distribution. A Z-score of 0 means the value is exactly at the mean, which in a normal distribution is the most probable single value.

Standard Deviation

Standard deviation is a measure that describes the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation implies that the values spread out over a wider range.
To calculate the standard deviation, you first determine the variance by averaging the squared deviations of each number from the mean. The standard deviation is then the square root of this variance.

• The formula for variance: \( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2 \)
• The formula for standard deviation: \( \sigma = \sqrt{\sigma^2} \)

Standard deviation serves as an essential tool in statistics, as it helps interpret data by providing context to the mean. It is fundamental when using Z-scores, as it gives the necessary scale by which we measure distances from the mean. Understanding standard deviation allows better insights into the reliability and variability of the data.

Mean

The mean, often referred to as the average, is a central measure of a dataset and provides a rough "center" point for the data. It is calculated by summing all the values in a dataset and then dividing by the number of values.
Mathematically, the mean is expressed as:

• \( \mu = \frac{1}{N} \sum_{i=1}^{N} X_i \)

where \( X_i \) represents each value in the dataset and \( N \) is the total number of values.
In statistical analysis, the mean offers a useful summary of the data, despite its sensitivity to extreme values (outliers). Knowing the mean is crucial when working with Z-scores, as it acts as the point from which deviations (measured in standard deviations) are calculated.
Overall, understanding the mean gives context to data, helping us appreciate not only the data's central tendency but also how individual data points relate to each other.