Question

The following table presents the annual personhours of time lost due to traffic congestion for a group of cities for 2007. This statistic is a measure of traffic congestion $$\begin{array}{lc} {\text { City }} & \text { Annual Person-Hours of Time Lost to Traffic Congestion per Year per Person } \\ \hline \text { Baltimore } & 25 \\ \text { Boston } & 22 \\ \text { Buffalo } & 5 \\ \text { Chicago } & 22 \\ \text { Cleveland } & 7 \\ \text { Dallas } & 32 \\ \text { Detroit } & 29 \\ \text { Houston } & 32 \\ \text { Kansas City } & 8 \\ \text { Los Angeles } & 38 \\ \text { Miami } & 27 \\ \text { Minneapolis } & 22 \\ \text { New Orleans } & 10 \\ \text { New York } & 21 \\ \text { Philadelphia } & 21 \\ \text { Pittsburgh } & 8 \\ \text { Phoenix } & 23 \\ \text { San Antonio } & 21 \\ \text { San Diego } & 29 \\ \text { San Francisco } & 29 \\ \text { Seattle } & 24 \\ \text { Washington, DC } & 31 \end{array}$$ a. Calculate the mean and median of this distribution. b. Compare the mean and median. Which is the higher value? Why? c. If you removed Los Angeles from this distribution and recalculated, what would happen to the mean? To the median? Why? d. Report the mean and median as you would in a formal research report.

Short answer

The mean is 22.8 and the median is 22.5. Without Los Angeles, the mean is 19.9, and the median is 22.

Step-by-step solution

- List All Values

First, list all the annual person-hours of time lost due to traffic congestion: \(25, 22, 5, 22, 7, 32, 29, 32, 8, 38, 27, 22, 10, 21, 21, 8, 23, 21, 29, 29, 24, 31\).

- Calculate the Mean

To find the mean, sum all the values and divide by the number of values. \[\text{Mean} = \frac{25 + 22 + 5 + 22 + 7 + 32 + 29 + 32 + 8 + 38 + 27 + 22 + 10 + 21 + 21 + 8 + 23 + 21 + 29 + 29 + 24 + 31}{22} = \frac{456}{22} = 22.8 \text{ person-hours} \]

- Calculate the Median

To find the median, list the values in ascending order and find the middle value. If the number of observations is even, the median is the average of the two middle values.Ordered list: \[5, 7, 8, 8, 10, 21, 21, 21, 22, 22, 22, 23, 24, 25, 27, 29, 29, 29, 31, 32, 32, 38\] As there are 22 data points, take the average of the 11th and 12th values: \[ \text{Median} = \frac{22 + 23}{2} = 22.5 \text{ person-hours} \]

- Compare Mean and Median

The mean (22.8) is higher than the median (22.5). The mean is affected by very high values such as 38 person-hours from Los Angeles, pulling it upward.

- Effect of Removing Los Angeles

Remove the value for Los Angeles (38) and recalculate.New sum: \[ 456 - 38 = 418 \]New mean: \[ \frac{418}{21} = 19.9 \text{ person-hours} \]For the median, we have an odd number of data points (21), so the median is now the 11th value in the ordered list: \[ 22 \text{ person-hours} \]

- Formal Report of Statistics

Before removing Los Angeles: - Mean annual person-hours of time lost: 22.8 - Median annual person-hours of time lost: 22.5 After removing Los Angeles: - Mean annual person-hours of time lost: 19.9 - Median annual person-hours of time lost: 22.

Key concepts

Mean Calculation

The mean, also known as the average, represents the central value of a data set.
It is calculated by summing all the individual observations and then dividing by the number of observations.
This provides a helpful insight into the overall level of traffic congestion across different cities.
In our exercise, we added all the person-hours of time lost due to traffic congestion and divided by the number of cities (22).
The formula used was:
\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{456}{22} = 22.8 \text{ person-hours} \]
This tells us that, on average, each person in these cities loses 22.8 hours annually due to traffic congestion.

Median Calculation

The median is another measure of central tendency.
It represents the middle value in a data set when the values are arranged in ascending order.
This gives us a better sense of the 'typical' level of traffic congestion.
If the number of observations is odd, the median is the middle value. For an even number of observations, the median is the average of the two middle values.
In our case, with 22 values, the middle values were the 11th and 12th entries in the ordered list.
The calculation was:
\[ \text{Median} = \frac{22 + 23}{2} = 22.5 \text{ person-hours} \]
This indicates that half the cities have congestion levels below 22.5 person-hours, and half have levels above.

Data Distribution Analysis

Data distribution gives us important information about how the congestion levels are spread across different cities.
It can help identify patterns, such as whether most cities cluster around the mean or if there are significant differences.
In our exercise, we saw that the values ranged from 5 to 38 person-hours.
Distributions can be visually represented using graphs like histograms or box plots.
For instance, seeing that many cities have values around the 20-25 hour mark tells us that moderate congestion is common.
This spread helps in understanding whether strategies to reduce traffic congestion are needed across all cities or only in specific instances.

Effect of Outliers

Outliers are extreme values that differ significantly from other observations.
They can majorly affect statistical measures like the mean.
In our dataset, Los Angeles with 38 person-hours is notably higher than the others.
When this outlier was removed, the new mean dropped from 22.8 to 19.9 person-hours.
The formula used was:
\[ \text{New Mean} = \frac{418}{21} = 19.9 \text{ person-hours} \]
However, the median was less affected, changing only from 22.5 to 22 person-hours.
This demonstrates that the median is more robust to outliers compared to the mean.

Formal Statistical Reporting

Accurate and clear statistical reporting is crucial for research and data communication.
It ensures that insights are understandable and actionable.
In a formal report, after calculating the mean and median, it’s important to present them clearly.
For example, before removing the outlier, the reported values were:
- Mean annual person-hours of time lost: 22.8
- Median annual person-hours of time lost: 22.5
After removing the outlier, the values were updated to:
- Mean annual person-hours of time lost: 19.9
- Median annual person-hours of time lost: 22
Such precise reporting helps decision-makers manage traffic congestion more effectively.