Question

A normal distribution has a mean of 20 and a standard deviation of \(4 .\) Find the \(Z\) scores for the following numbers: (a) 28 (b) 18 (c) 10 (d) 23.

Short answer

Z-scores: (a) 2, (b) -0.5, (c) -2.5, (d) 0.75.

Step-by-step solution

Understanding Z-score Calculation

The Z-score formula is \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value from the data set, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.

Calculate Z-score for 28

Substitute \( X = 28 \), \( \mu = 20 \), and \( \sigma = 4 \) into the Z-score formula: \[ Z = \frac{28 - 20}{4} = \frac{8}{4} = 2 \].

Calculate Z-score for 18

Substitute \( X = 18 \), \( \mu = 20 \), and \( \sigma = 4 \) into the Z-score formula: \[ Z = \frac{18 - 20}{4} = \frac{-2}{4} = -0.5 \].

Calculate Z-score for 10

Substitute \( X = 10 \), \( \mu = 20 \), and \( \sigma = 4 \) into the Z-score formula: \[ Z = \frac{10 - 20}{4} = \frac{-10}{4} = -2.5 \].

Calculate Z-score for 23

Substitute \( X = 23 \), \( \mu = 20 \), and \( \sigma = 4 \) into the Z-score formula: \[ Z = \frac{23 - 20}{4} = \frac{3}{4} = 0.75 \].

Key concepts

Normal distribution

In statistics, a normal distribution is a highly important concept. It's often called the bell curve due to its iconic shape when the data is plotted on a graph. This type of distribution helps to understand how data points tend to be distributed around the mean.
Normal distributions are symmetrical, meaning most data points cluster around the mean, decreasing gradually as you move away. The mean, median, and mode of a normal distribution are all the same. This symmetry and bell shape allows statisticians to make inferences about a population.
In real-world scenarios, many things follow a normal distribution: heights of people, test scores, or any natural phenomenon where outcomes are influenceable by countless little factors. Seeing how the mean and standard deviation affect this curve helps in predicting probabilities of various outcomes.

Mean and standard deviation

The mean and standard deviation are two critical statistics that give insight into the data being analyzed. The mean is the average of all data points, offering a central value around which the data clusters. Calculating the mean is simple:

• Add together all the data points.
• Divide the sum by the number of data points.

The standard deviation, on the other hand, measures the amount of variability or spread in a set of data. A small standard deviation means that most of the numbers are close to the mean, whereas a large standard deviation indicates that the data points are spread over a broader range of values.
Understanding these concepts is crucial when discussing a normal distribution because they control the shape and spread of our bell curve. A standard deviation can significantly alter how data speaks, influencing decisions in research, business, and more.

Statistical formulas

Statistical formulas are mathematical equations used to calculate different statistics, which help us make sense of data. They offer a systematic way to compute values that summarize complex data sets in simple terms.
For example, to find a Z-score, which tells us how far a particular data point is from the mean in terms of standard deviations, we use the formula:

• \[ Z = \frac{X - \mu}{\sigma} \]

Here, \(X\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Z-scores are a type of standard score that indicates the position of a score in relation to the mean of a group of scores.
Using statistical formulas like this allows us to standardize different scales or measurements into a normal distribution, thus facilitating easier comparison and ultimately more informed decision-making.