Problem 14 - Algebra 2

Question

Identify the outlier of each set of values. Then describe how its value affects the mean of the data. $$ 947 \quad 757 \quad 103 \quad 619 \quad 661 \quad 582 \quad 626 \quad 900 \quad 869 \quad 728 \quad 1001 \quad 596 \quad 515 $$

Step-by-Step Solution

Answer

The outlier in the provided set of values is 103. Including this outlier in the calculation of the mean reduces the mean from approximately 742.25 to 692.31.

Step 1: Identifying the outlier

Looking at our data set, the number 103 stands out, as it is much smaller than the other numbers. So, the number 103 can be considered as the outlier in this set of numbers.

Step 2: Calculating the mean with the outlier

The mean (also known as average) of a set of numbers is obtained by adding up all the numbers and then dividing by the total count. So, adding up all the numbers in our set, including the outlier, we get 9010. There are 13 numbers, so the mean with the outlier is \( \frac{9010}{13} \)\approx 692.31.

Step 3: Calculating the mean without the outlier

To calculate the mean without the outlier, we subtract the outlier (103) from the total sum (9010), giving 8907. As this leaves 12 numbers, the mean without the outlier is \( \frac{8907}{12} \)\approx 742.25.

Step 4: Describing the effect of the outlier

Comparing the two means from previous steps, it is evident that when the outlier (103) is included, it significantly lowers the mean. In this case, it decreases the mean from approximately 742.25 to 692.31. Thus, the outlier in this data set has a noticeable lowering effect on the mean value of the data.

Key Concepts

Mean Calculation

The mean, often referred to as the average, is a fundamental statistical measure. It provides us with a central value around which a data set is dispersed. To calculate the mean, you follow a simple process:
Add up all the numbers in your data set to find their total sum.
Count how many numbers are in the data set.
Divide the total sum by the count of numbers.
This formula can be mathematically expressed as:

• Mean = \( \frac{\text{Sum of all values}}{\text{Number of values}} \)

In the context of our exercise, by including all 13 numbers, the total was 9010, which when divided by 13, gives a mean of approximately 692.31. This is the mean calculated when the entire data set is considered.

Effect on Mean

Outliers in data can heavily influence the mean, sometimes giving a false impression of the central tendency. In this exercise, we observed the shift in the mean value when the outlier was included versus when it was excluded. When the outlier, which is a significantly smaller value (103), is part of the data set:

• The total sum decreases since the outlier is substantially lower than the rest of the numbers.
• This reduction results in a lower calculated mean, pulling it away from what may be considered the 'true' central value for the other numbers.

The calculated mean with the outlier was approximately 692.31. Without the outlier, the mean increased to approximately 742.25, reflecting a truer central tendency of the rest of the data set.
This example highlights how outliers can distort the mean, making it essential to recognize and understand their impact.

Identifying Outliers

Outliers are values within a data set that deviate markedly from the others. These can arise due to measurement variability, errors, or genuine anomalies. Identifying them is crucial for accurate data analysis:

• Visually scan the list of numbers to spot values that are unusually high or low.
• Calculate the potential impact by examining how these affect statistical measures like the mean.

In the given example, 103 stood out because it was notably smaller than all other values, making it an outlier.
Recognizing outliers helps in deciding whether they reflect actual variability or an anomaly that should be taken into account or discarded, depending on the situation.