5.2: Time Plots & Measures of Central Tendency
Learning Objectives
- Construct time plots
- Describe trends in time plots
- Calculate range and measures of central tendency: mean, median, mode
- Understand how a change in the data will effect the statistics
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.
Year |
Municipal Waste Generated (Millions of Tons) |
1990 | 269 |
1991 | 294 |
1992 | 281 |
1993 | 292 |
1994 | 307 |
1995 | 323 |
1996 | 327 |
1997 | 327 |
1998 | 340 |
Solution
a) In this example, the time (in years) is considered the explanatory variable, and is graphed along the horizontal axis. The amount of municipal waste is the response variable, and is graphed along the vertical axis. Time plots can be drawn by hand, graph paper makes this easier, or created with computer software programs, or graphing calculators. This example was made using Excel.
b) This graph shows that the amount of municipal waste generated in the United States increased at a fairly steady rate during the 1990s. Between 1991 and 1992 there was a decrease of 13 million tons of municipal waste, but every other year during the 1990s had an increase.
c) It should be noted that factors other than the passage of time cause our waste to increase. Other factors, such as population growth, economic conditions, and societal habits and attitudes also contribute as causes.
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?
Source.http://mste.illinois.edu Aug 1,2011.
Solution
a) The greatest increase appears to have happened during the 1990s, when it went from ≈$3.75 to ≈$5.20.
b) The 1970s appear to have had 5 or 6 increases in the minimum wage.
c) The longest constant minimum wage was during the 1980s.
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.
Example: Suppose these are the hourly wages for the employees at Burger Boy: $7.25, $7.55, $8.15, $7.40, $7.25, $8.90, $16.75, $8.10. If you calculate the mean wage, you get $8.92. So, if someone were to report the average wage at Burger Boy to be $8.92 it would give the impression that this is what the average employee makes. However, this is misleading because everyone other than the manager makes less than this amount. So, the mean is very misleading in this case. The manager's higher salary is causing the mean to be higher.
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.
Example: If you calculate the median salary for the Burger Boy employees you get $7.83. This is a much better description of what the typical employee at Burger Boy gets paid because half the employees make more than this amount and half make less than this amount. The manager's higher salary does not affect the median.
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:
\begin{align*}24, 24, 31, 50, 53, 66, 78\end{align*}
What is the mean number of hours that Stephen worked per month?
Solution
Stephen has worked at Wendy’s for 15 months but the numbers given above are for seven months. Therefore, this set of data represents a sample of the population. The formula that is used to calculate the mean for a sample and for a population is the same. However, the symbols are different. The mean of a sample is denoted by \begin{align*}\bar{x}\end{align*} which is called “\begin{align*}x\end{align*} bar”. The mean of an entire population is denoted by \begin{align*}\mu\end{align*} which is the Greek letter "mu" (pronounced "myoo").
The number of data for a sample is written as \begin{align*}n\end{align*}. The following formula represents the steps that are involved in calculating the mean of a sample:
\begin{align*}Mean = \frac{add \ the \ numbers}{the \ number \ of \ numbers}\end{align*}
This formula can now be written using symbols.
\begin{align*}\overline{x}=\frac{x_1+x_2+x_3+ \ldots + x_n}{n}\end{align*}
You can now use the formula to calculate the mean of the hours that Stephen worked.
\begin{align*}\overline{x} &= \frac{ x_1+x_2+x_3+ \ldots + x_n}{n}\\ \overline{x} &= \frac{24+25+33+50+53+66+78}{7}\\ \overline{x} &= \frac{329}{7}\\ \overline{x} &= 47\end{align*}
The mean number of hours that Stephen worked during this time period was 47 hours 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.
\begin{align*}23, 21, 29, 24, 31, 21, 27, 23, 24, 32, 33, 19\end{align*}
Solution
mean: \begin{align*}\left ( 23+21+29+24+31+21+27+23+24+32+33+19\right )/12=307/12\end{align*}
\begin{align*}307/12=25.58\end{align*}
median: first, organize the ages in ascending order \begin{align*}19, 21, 21, 23, 23, 24, 24, 27, 29, 31, 32, 33\end{align*}
second, count in to find the middle value \begin{align*}24, 24\end{align*} the middle value will be halfway between these two values (or the average of these two values) \begin{align*}\frac{24+24}{2}=24\end{align*}
mode: look for the value(s) that occur most frequently \begin{align*}21, 23, 24\end{align*} this data set has three modes
range: subtract the smallest value from the largest value (max - min = range) \begin{align*}33-19=14\end{align*}
Solution: make your conclusion in context
At this coffee shop, the mean age of people in this sample is 25.58 years old and the median age is 24 years old. There were three modes for age at 21, 23, and 24 yeas old and the range for ages is 14 years.
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.
a) What is the lowest grade Lulu will need to get on the last test in order to get an A in health?
b) What is the lowest grade Lulu will need to get on the last test in order to get an A- in health?
Solution
a) So she sets up an equation thinking about how she would calculate her average test grade if she knew all six scores. Knowing that she wants the final average to equal 93%, she puts an 'x' in the place of the last test score, and then does some algebra to solve for x.
\begin{align*}\frac{85+95+77+89+94+x}{6}=93\end{align*}
\begin{align*}\left ( 85+95+77+89+94+x\right )=93*6\end{align*}
\begin{align*} 85+95+77+89+94+x=558\end{align*}
\begin{align*}440+x=558\end{align*}
\begin{align*}x=118\end{align*}
Oh no! There is no way she can get 118%. So, there is no possible hope for her to get an A.
b) It is time to try for an A-, but that 118% scared her, so she is going to think of the lowest possible score that will still be an A-. With rounding, if she can get her mean score to 89.5%, she will make it. So she tries the same algebra, but with 89.5 as the final result.
\begin{align*}\frac{85+95+77+89+94+x}{6}=89.5\end{align*}
\begin{align*}\left ( 440+x\right )=89.5*6\end{align*}
\begin{align*}440+x=537 \end{align*}
\begin{align*}x=97\end{align*}
There is hope! As long as she gets a 97% or higher on this last test, she can get an A-. She is going to study like crazy!
Problem Set 5.2
Section 5.2 Exercises
1) Determine the mean, median, mode and range for each of the following sets of numbers:
a) 20, 14, 54, 16, 38, 64
b) 22, 51, 64, 76, 29, 22, 48
c) 40, 61, 95, 79, 9, 50, 80, 63, 109, 42
2) The mean weight of five men is 167.2 pounds. The weights of four of the men are 158.4 pounds, 162.8 pounds, 165 pounds and 178.2 pounds. What is the weight of the fifth man?
3) The mean height of 12 boys is 5.1 feet. The mean height of 8 girls is 4.8 feet.
a) What is the total height of the boys?
b) What is the total height of the girls?
c) What is the mean height of the 20 boys and girls all together?
4) The following data represents the number of advertisements received by ten families during the past month. Make a statement describing the 'typical' number of advertisements received by each family during the month. Be sure to include statistics to support your statement.
\begin{align*}43 \qquad 37 \qquad 35 \qquad 30 \qquad 41 \qquad 23 \qquad 33 \qquad 31 \qquad 16 \qquad 21\end{align*}
5) Mica's chemistry teacher bases grades on the average of each student's test scores during the trimester. Mica has been kind of slacking this year, but hasn't been too concerned because he knows that he will at least get the credit (60% = passing). However, his parents just informed him that he will not be allowed to use the car if he has any grade below a C (73%). Here are Mica's chemistry test scores for the first eight chapters:
\begin{align*}10, 70, 71, 82, 65, 76, 58, 75\end{align*}
a) Calculate the mean, median, mode, and range for Mica's chemistry tests. What grade will Mica receive in chemistry based on this?
b) His teacher has decided that each student may retake any one of his or her tests in an effort to improve his or her grade. Mica jumps at this opportunity, studies chapter one for hours and retakes the test. To his, and his mother's delight, his 10% turns into a 70%!! Woo-hoo! Calculate the mean, median, mode, and range for Mica after this change. Which of these values changed? Which did not? What grade will Mica receive now?
c) If Mica continues to study and earns a 60% on the chapter 9 test and a 76% on the chapter 10 test, what will his final average be?
d) If Mica continues to study and earns an 85% on the chapter 9 test and a 90% on the chapter 10 test, what will his final average be?
6) Deals on Wheels: The following table lists the retail price and the dealer’s costs for 10 cars at a local car lot this past year:
Car Model | Retail Price | Dealer’s Cost |
Amount of Mark-Up |
Percent of Mark-Up |
Nissan Sentra | $24,500 | $18,750 | ||
Ford Fusion | $26,450 | $21,300 | ||
Hyundai Elantra | $22,660 | $19,900 | ||
Chevrolet Malibu | $25,200 | $22,100 | ||
Pontiac Sunfire | $16,725 | $14,225 | ||
Mazda 5 | $27,600 | $22,150 | ||
Toyota Corolla | $14,280 | $13,000 | ||
Honda Accord | $28,500 | $25,370 | ||
Volkswagen Jetta | $29,700 | $27,350 | ||
Subaru Outback | $32,450 | $28,775 |
a) Calculate the amount each car was marked up.
b) Calculate the percent that each car was marked up \begin{align*}\frac{mark-up}{dealer-cost}*100\%=\end{align*} Report answers rounded to the nearest one-tenth of a percent.
c) Calculate the mean, median, mode and range for the percent of mark-up.
d) Do the "amount of mark-up column" and the "percent of mark-up column" put the cars in the same order for profit? Explain or give an example.
7) Write a brief description of what the line graph for Platinum Prices shows. Be sure that you do this in context, as complete sentences, and that you include at least three observations.
Source.http://www.admc.hct.ac.ae
8) According to the U.S. Census Bureau, "household median income" is defined as "the amount which divides the income distribution into two equal groups, half having income above that amount, and half having income below that amount." The table shows the median household income data, every 3 years, from 1975 until 2008, according to the U.S. Census Bureau.
a) Construct a time plot for the median household data. You may do this by hand on graph paper, or by using technology.
b) Write a brief description of what the line plot shows. This should be done as complete sentences, in context of the distribution, and should include at least three distinct observations.
Review Exercises
For each of the following problems, decide whether you will use a combination, a permutation, or the fundamental counting principle. Then, set up and solve the problem.
9) A camp counselor is in charge of 10 campers. The kids will be going horseback riding today. There are 5 horses, so they will go in two shifts. In how many ways can the camp counselor assign campers to the specific horses for the first shift?
10) In how many ways can the camp counselor select 4 campers, out of the ten, to attend the afternoon archery class?
11) How many pizzas are possible, made of three different toppings, when 12 toppings are available to select from?
12) Luigi has 3 pairs of shoes, 7 pairs of jeans, and 8 shirts that he likes to wear and that are clean. He is going to put together an outfit for his hot date tonight. If he will choose one of each, how many outfits are possible?
13) Eleven skiers are to be in a race. Prizes will be awarded for 1st, 2nd and 3rd places. Assuming no ties, in how many ways might the prizes be awarded?
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