5.2 Notes/Examples

5.2 Summary Notes

Line Graphs as Time Plots

We are often interested in how something has changed over time. The type of graphical display that shows this the most clearly is a time plot, or line graph. When one of the variables is time, it will almost always be plotted along the horizontal axis (as the explanatory variable). Because time is a continuous variable and we are trying to see if there is some type of trend in how the other variable (response) has behaved over a period of time, a line graph is often very useful in showing this relationship. Source: http://www.zerowasteamerica.org

Example 1

The total municipal waste generated in the US by year is shown in the following data set.

a) Construct a time plot to show the change in the amount of municipal waste generated in the United States during the 1990's.

b) Comment on the trend that is shown in the graph.

c) Suggest factors (other than time) that may be leading to this trend.

Example 2

Here is a line graph that shows how the hourly minimum wage changed from when it was first mandated through 1999.

a) During which decade did the hourly wage increase by the greatest amount?

b) During which decade did it increase the most times?

c) When did it stay constant for the longest?

[Figure3]

Source.http://mste.illinois.edu Aug 1,2011.

Solution

Measures of Central Tendency & Spread

The mean, the median, and the mode are all measures of central tendency. They all show where the center of a set of data "tends" to be. Each one is useful at different times. Any one of these three measures of central tendency may be referred to as the average of a set of data.

Mean

The mean, often called the ‘average’ of a numerical set of data, is the sum of all of the numbers divided by the number of values in the data set. This value is the arithmetic mean, and it tells us what value we would have if all of the data were the same. The mean is the balance point of a distribution, and is one of the three measure of central tendency commonly used in statistics. The mean is a summary statistic that gives you a description of the entire data set and is especially useful with large data sets where you might not have the time to examine every single value. However, the mean is affected by extreme values, called outliers, and can end up leaving the observer with the wrong impression of a data set.

Median

The median is the number in the middle position once the data has been organized. Organized data is simply the numbers arranged from smallest to largest or from largest to smallest. This is the only number for which there are as many above it as below it in the set of organized data, and is referred to as the equal areas point. The median, for an odd number of data, is the value that is exactly in the middle of the ordered list, it divides the data into two halves. The median for an even number of data, is the mean of the two values in the middle of the ordered list. The median is useful when there are a few extreme values that can effect the mean, because the middle number will stay in the middle. The median often gives a good impression of the center, because there are 50% of the values above the median, 50% of the values below the median, and it doesn't matter how big the biggest values are or how small the smallest values are.

Mode

The mode of a set of data is simply the number that appears most frequently in the set. There are no calculations required to find the mode of a data set. You simply need to look for it. However, be aware that it is common for a set of data to have no mode, one mode, two modes or more than two modes. If there is more than one mode, simply list them all. And, if there is no mode, write 'no mode'. No matter how many modes, the same set of data will have only one mean and only one median. The mode is a measure of central tendency that is simple to locate but is not used much in practical applications. It is the only one of these three values that can be for either categorical or numerical data. Remember the example regarding pets? The mode was 'dogs' because that was the most common response.

Range

The range of a data set describes how spread out the data is. To calculate the range, subtract the smallest value from the largest value (maximum value -- minimum value = range). This value provides information about a data set that we cannot see from only the mean, median, or mode. For example, two students may both have a quiz average of 75%, but one of them may have scores ranging from 70% to 82% while the other may have scores ranging from 24% to 90%. In a case such as this, the mean would make the students appear to be achieving at the same level, when in reality one of them is much more consistent than the other.

Example 3

Stephen has been working at Wendy’s for 15 months. The following numbers are the number of hours that Stephen worked at Wendy’s during the past seven months:

What is the mean number of hours that Stephen worked per month?

Example 4

The ages of several randomly selected customers at a coffee shop were recorded. Calculate the mean, median, mode, and range for this data.

Example 5

Lulu is obsessing over her grade in health class. She just simply cannot get anything lower than an A-, or she will cry! She knows that the grade will be based on her average (mean) test grade and that there will be a total of six tests. They have taken five so far, and she has received 85%, 95%, 77%, 89%, and 94% on those five tests. The third test did not go well, and she is getting worried. The cutoff score for an A is 93%, and 90% is the cuttoff score for an A-. She wants to know what she has to get on the last test. The teacher assures her that she will round to the nearest whole percent.

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