Statistics and Probability-Part 2

There are several statistical measures that can be used to compare two data sets. Measures of Central Tendency: Mean is the average of the data set. The Mean can be found by adding up the data, and dividing by the number of “elements” in the data set. Median is the middle term in a data set. This can be found by re-ordering the numbers from least to greatest, then finding the middle number. If there are two “middle numbers”, the median can be found by taking the average of those two numbers. Mode is the element in the data set that occurs the most often.

Measures of Variability: It is also important to consider the spread of a data set in order to make accurate claims. Another way to define variability is the “spread” of the data, or distance between values. Variability can be measured using range, inter-quartile range, and mean absolute deviation. Range is the difference between the maximum value and the minimum value in the data set. Mean Absolute Deviation can be found by listing the absolute value of how far each data point is away from the mean, and calculating the average of the list. This shows how “spread” the data is.

How Much Hotter? Jerome lives in Minneapolis and is visiting his cousin Evelyn in Denver during his summer vacation. Jerome complains to Evelyn that it is much hotter in Denver than in Minneapolis, but Evelyn thinks that the weather in Denver is not very different from the weather in Minneapolis during the summer. They decide to investigate by reviewing the previous July’s daily high temperatures in both cities. Their findings are shown below.

After reviewing the data, both Jerome and Evelyn thought they were correct. What observations can you make about the data? They both decided that finding the mean, or arithmetic average, of each data set would be helpful. Jerome thought the mode, or temperature that appears the most often, could be helpful as well. Evelyn wanted to find the median, or middle temperature of the data set, once she listed them in order to help her prove her claim. Find the following measures of central tendency for the high temperature data for each city, to the nearest degree.

Mean: Median: Mode:

Based on what you found in the problem above, who do you think is right? Do you think that July in Denver is much hotter than July in Minneapolis? Explain. Which measure of central tendency was the most helpful for you in making your decision? Which was least helpful?

Measuring Variability Jerome thinks that the range in the data set will prove that Denver is “much hotter” than Minneapolis in July. Calculate the range of the daily high temperatures for Denver and for Minneapolis. Do these numbers help Jerome support his claim? Explain your reasoning.

Evelyn reviewed the ranges and can’t decide if this helps her prove her claim or not. She thinks the range along with the other numbers she calculated will be helpful. Considering the ranges and other measures of central tendency that you calculated, do you think the difference in the July temperatures of the two cities is small or large? Explain. To get a more detailed image of how the data is spread across the range, we can make a dot plot of the data. Below are dot plots created using the same scale for the daily high temperatures in July for Denver and Minneapolis.

Examine the two dot plots above and compare the variability of the data for the two cities. Make some observations about the spread of each city’s data. Be sure to include information about clustering of data around certain temperatures. Also, identify any areas of overlap between the two cities. Jerome is convinced that the dot plots prove his claim, even if the ranges of the two data sets are similar. Explain what you think Jerome sees in the dot plots that the range alone does not show. Would you expect the mean to be in the middle of the data plots? What do you notice in regard to the location of the mean as compared to the spread of the data. Explain.