Problem 28 - Algebra 2

Question

CHALLENGE The graphing calculator screen shows the graph of a normal distribution for a large set of test scores whose mean is 500 and whose standard deviation is \(100 .\) If every test score in the data set were increased by 25 points, describe how the mean, standard deviation, and graph of the data would change.

Step-by-Step Solution

Answer

The mean increases to 525, standard deviation stays 100, and the graph shifts right by 25.

Step 1: Understanding the Problem

We are given a normal distribution with a mean (\(\mu\)) of 500 and a standard deviation (\(\sigma\)) of 100. We need to determine how these change if each test score is increased by 25 points.

Step 2: Effect on the Mean

When every score in a data set is increased by the same amount, the mean of the data set increases by that same amount. Therefore, if each score is increased by 25, the new mean becomes \(500 + 25 = 525\).

Step 3: Effect on the Standard Deviation

The standard deviation measures the spread of the data around the mean. When every score is increased by a constant amount, the spread of the data does not change. Hence, the standard deviation remains \(100\).

Step 4: Effect on the Graph

The graph of a normal distribution is symmetric about the mean. Increasing every score by 25 shifts the entire distribution to the right by 25 units, but does not change the shape or spread of the graph.

Key Concepts

Understanding the Mean in Normal Distribution

The mean in a normal distribution is a fundamental concept in statistics. It is essentially the average of all values in a data set. In mathematical terms, the mean is denoted by the Greek letter \( \mu \). In the context of a normal distribution, the mean is the point at which the graph is perfectly symmetrical. It divides the graph into two equal halves.
For instance, if we were given a mean of 500 for a large set of test scores, it signifies that the average score is 500. Now, suppose every score increases by 25 points. The impact on the mean would be a direct increase by that same amount. Thus, the new mean would be \(500 + 25 = 525\).
Essentially, adding a constant to each data point results in the mean also increasing by that constant amount. This shift helps maintain the symmetry of the distribution graph about its central axis.

Understanding Standard Deviation

Standard deviation is a crucial statistic used to measure the variability or spread of a data set. It is symbolized by \( \sigma \) and tells us how much the values in a set deviate from the mean. A smaller standard deviation implies that the values are closely clustered around the mean, while a larger standard deviation indicates a wider spread.
In the exercise, our data set initially has a standard deviation of 100. This means that most test scores would fall within 100 points of the mean. However, if we increase each score by 25, the standard deviation remains unchanged at 100.
Why does the standard deviation remain the same? It’s because adding a constant to each score does not alter the relative positioning or the spread of the scores around the mean. The shift affects the mean itself, but not the spread or variability of the data.

Graph Transformation in Normal Distribution

When discussing graph transformations, particularly in a normal distribution, we focus on the shape and positioning of the bell curve graph. The "bell curve" characterizes normal distribution graphs, which are symmetric and centered around the mean.
In moving every test score up by 25 points, we effectively shift the entire graph 25 units to the right. This type of transformation is termed a "horizontal shift". The overall shape—still resembling a bell with the same spread—does not alter.
Such transformations only affect the graph's position, not its spread, because each score is adjusted equally. Thus, while the mean changes, causing the graph's center to shift rightwards, the graph maintains its original form, with its symmetry and standard deviation unaltered. This preservation of shape is a hallmark of consistent transformations across a data set.