Problem 16 - Algebra 2

Question

Writing In a class of \(25,\) one student receives a grade of 100 on a test. The grades are distributed approximately normally, with a mean of 78 and a standard deviation of \(5 .\) Do you think the student's grade is an outlier? Explain.

Step-by-Step Solution

Answer

Yes, the student's grade of 100 is an outlier, as it is 4.4 standard deviations away from the mean, which is significantly higher than the common threshold of 2 standard deviations away from the mean.

Step 1: Identify and Understand the Data

For this exercise, the mean grade is 78, the standard deviation is 5, and the student's grade is 100.

Step 2: Compute the z-score

The formula for z-score is \(Z = (X - μ) / σ\), where X is the data point (the student’s score), μ (mu) is the mean, and σ (sigma) is the standard deviation. So, the z-score can be computed as follows: \(Z = (100 - 78) / 5 = 22 / 5 = 4.4\). This means that the student's score is 4.4 standard deviations away from the mean.

Step 3: Interpret the z-score

A z-score of 4.4 is significantly larger than 2, which is a common threshold that indicates that data point is significantly different than the mean and could therefore be considered an outlier. Moreover, in a normally distributed data set, only about 5% of the values fall more than 2 Standard Deviations from the mean. Therefore, a z-score of 4.4 indicates that the student's score is very unusual.

Key Concepts

Standard Deviation

The standard deviation is a measure used to quantify the amount of variation or dispersion in a set of data values. It tells us how much the individual data points deviate from the mean, or average, of the dataset.
When the standard deviation is small, the data points tend to be close to the mean. Conversely, a large standard deviation indicates wider dispersion of data points around the mean.
Understanding standard deviation is crucial because it provides insights into the variability of the dataset:

• A lower standard deviation indicates that data points are generally similar and are clustered closely around the mean.
• A higher standard deviation signifies that data points are spread out over a larger range of values.
• With this in mind, in our example, the standard deviation is 5, indicating that most of the students’ grades are relatively close to the mean grade of 78.

Being able to interpret the standard deviation effectively helps in assessing the spread and consistency of the data being analyzed.

Normal Distribution

The concept of normal distribution, often referred to as the bell curve, is a fundamental idea in statistics. A normal distribution is a continuous probability distribution that is symmetrical on both sides of the mean, showcasing how data is naturally spread around the central value.
Normal distributions have distinct characteristics:

• The mean, median, and mode of a normally distributed data set are all equal.
• The curve is bell-shaped and symmetrical about the mean, meaning most of the data points lie close to it.
• As we move away from the mean, the probability of data points decreases. Extreme values are less likely to occur.

In the context of our exercise, the grades are said to be "approximately normally distributed." This implies that most students score close to the average score of 78, while very high or very low scores are less common. Grades show a natural tendency to cluster around the mean, with fewer students achieving much higher or lower scores.

Mean

The mean, also known as the average, is a measure of central tendency that is calculated by adding up all the data points and dividing by the number of values. It provides a simple summary of a dataset and is essential in many statistical analyses.
To find the mean, use the formula: \[\text{Mean (μ)} = \frac{\sum X}{N}\]where \(\sum X\) is the sum of all data values, and \(N\) is the number of data points.
The mean is crucial for identifying how data is distributed around a central value:

• In our exercise, the mean is 78. This is the point around which most students' grades are centered.
• The mean provides a reference point to which we can compare other values, such as calculating the z-score in the exercise.
• It is useful in determining what might be considered an outlier, or an unusual data point, based on its distance from the mean.

Understanding the mean helps in assessing the general performance trend of a dataset, providing context for interpreting individual scores relative to the overall distribution.