Problem 1 - Algebra 2 and Trigonometry

Question

A student's scores on five tests were \(98,97,95,93,\) and \(67 .\) Explain why this set of scores does not represent a normal distribution.

Step-by-Step Solution

Answer

The data set has an outlier (67) causing asymmetry, hence it's not a normal distribution.

Step 1: Define a Normal Distribution

A normal distribution, also known as a Gaussian distribution, is a type of continuous probability distribution for a real-valued random variable. Its graph is bell-shaped, symmetric about the mean, and characterized by the mean, median, and mode being equal.

Step 2: Review the Data Set

The given scores are 98, 97, 95, 93, and 67. On first glance, most of the scores are quite close to each other (98, 97, 95, 93) except for the score of 67, which is significantly lower than the others.

Step 3: Check for Symmetry and Uniform Spread

For a distribution to be normal, the data points should be symmetrically distributed around the mean. These scores are not spread symmetrically; instead, there is a significant drop at 67, creating a tail on one side.

Step 4: Analyze the Impact of the Outlier

The score of 67 is an outlier because it is much lower than the rest of the scores. Since a normal distribution has roughly equal numbers of data points on either side of the central value (mean), this asymmetrical distribution caused by the outlier disrupts the symmetry.

Step 5: Conclusion

Since a normal distribution requires a symmetrical shape around the mean and no significant outliers, this data set cannot be a normal distribution due to the outlier score of 67 causing asymmetry.

Key Concepts

Understanding Outliers

An outlier is a data point that is distinctively different from the other data points in a dataset. In the context of a normal distribution, which requires data to be symmetrically arranged around the mean, an outlier can skew results.
In our example, the outlier is the score of 67. While the other scores—98, 97, 95, and 93—are closely packed together, 67 stands out as being markedly lower.

• This single outlier can have a significant impact on statistical measures, such as the mean, potentially misleading the interpretation of data.
• Outliers can occur due to variability in the measurement or possibly indicate an experimental error. Sometimes they provide useful information about variability in the process or population being studied.

The Importance of Symmetry in Distributions

Symmetry in data distribution refers to how numbers are spread in relation to the mean. For a distribution to be classified as normal, the data should form a symmetric shape.
In a normal distribution, the left and right sides of the graph should be mirror images of each other.

• This symmetry suggests that there is a similar pattern of distribution both to the left and right of the center (mean).
• When data isn't symmetrical, like in our example where the presence of 67 pulls the distribution out of balance, it disrupts the bell-shaped curve expected in a normal distribution.

The Gaussian Distribution

The Gaussian distribution, also known popularly as the normal distribution, is one of the most important probability distributions in statistics. It's known for its bell-shaped curve. This characteristic shape is equally distributed around the mean.
The Gaussian distribution is defined by its mean and standard deviation, where:

• The mean determines the center of the distribution.
• The standard deviation describes the width of this curve.

For a dataset to be normally distributed:

• The graph of its frequency should form a symmetric bell curve.
• There should be no anomalous deviations that create tails on either side of the curve.
• The skew of such deviations demonstrates misalignment with a Gaussian distribution, as seen in our dataset due to the score of 67.

Mean, Median, and Mode Equality

In a perfectly normal or Gaussian distribution, the mean, median, and mode coincide—they are all equal. This peculiar alignment indicates that the data is balanced around the central peak.

• The mean is the average of all data points.
• The median is the middle value when data points are ordered.
• The mode is the value that appears most frequently.

In a normal distribution:

• Each of these measures will produce the same number.
• However, in our example, the presence of a significant outlier (67) skews the mean, which drastically diminishes the probability that all three statistical measures align.

Thus, the score of 67 disrupts the balance and eradicates the equality of mean, median, and mode, which is essential for a normal distribution.