Problem 14 - Algebra 2

Question

A data set has mean 25 and standard deviation \(5 .\) Find the \(z\) -score of each value. $$ 11 $$

Step-by-Step Solution

Answer

The z-score for the given value is \(Z = -2.8\).

Step 1: Understand the Z-score formula.

The formula to calculate the z-score is (X - μ) / σ. Here, X is the data point, μ is the mean and σ is the standard deviation.

Step 2: Substitute the values

Substitute the given values into the formula. So, \(Z = (11 - 25) / 5\).

Step 3: Calculate the z-score

By solving the equation above, \(Z = -14 / 5 = -2.8\).

Key Concepts

Statistics

Statistics is a branch of mathematics that deals with collecting, analyzing, interpreting, and presenting data. It plays a crucial role in various fields such as economics, medicine, and social sciences.
Understanding statistics helps in making informed decisions based on data. There are two main types of statistics: descriptive and inferential.

• **Descriptive statistics** summarize and present data in a meaningful way using numbers, such as mean, median, mode, and standard deviation.
• **Inferential statistics** allow us to make predictions or inferences about a population based on a sample of data.

Knowing the basics of statistics, like mean and standard deviation, is essential for understanding more complex topics like z-scores, which are used to determine how far away a particular data point is from the mean.

Mean

The mean, often called the average, is a measure of central tendency that summarizes a set of data by using a single number.
To calculate the mean, you sum up all the values in a data set and then divide by the number of values. For example, if you have a data set of 5 numbers: 2, 4, 6, 8, and 10, the mean is calculated as:
\[ \text{Mean} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6 \]The mean provides a simple way to describe the center of the data. However, it can be affected by extreme values, called outliers.
In the provided exercise, the mean is 25, meaning that the average of all values in the data set is 25.

Standard Deviation

The standard deviation is a measure of the amount of variation or spread 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 are spread out over a wider range.
To calculate the standard deviation, you need to follow these steps:

• Find the mean of the data set.
• Subtract the mean from each value to get the deviation of each observation.
• Square each deviation to eliminate negative numbers.
• Find the average of these squared deviations.
• Take the square root of this average, which is the standard deviation.

In the exercise, the standard deviation is 5, meaning that the typical deviation from the mean (25) is 5 units. Understanding standard deviation is crucial for z-score calculations because it helps us understand how far a particular value is from the mean in relation to other values.