8.2: Frequency Distributions (Grouped Data)
- Organize data for frequency tables.
- Create and interpret time series graphs and scatter plots.
Frequency tables
Frequency tables can be used to summarize a dataset. You will create frequency tables in class. Here, we will practice organizing the data we need to make a frequency table. Begin constructing a frequency table by tallying each observation and then recording the frequency.
In a survey, 20 mothers were asked how many times per week a teenager must be reminded to do his or her chores. Here are the results:
3 5 2 4 4 1 4 3 2 5 2 3 1 2 4 0 4 3 2 5
| Number of times teenager is reminded | Tally | Frequency |
|---|---|---|
| 0 | | | 1 |
| 1 | || | 2 |
| 2 | ||||| | 5 |
| 3 | |||| | 4 |
| 4 | ||||| | 5 |
| 5 | ||| | 3 |
In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in Table. Construct a line graph.
| Number of times in shop | Frequency |
|---|---|
| 0 | 7 |
| 1 | 10 |
| 2 | 14 |
| 3 | 9 |
Answer
Time Series Graphs
Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.
One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don't have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.
To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.
The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States.
| Ukraine | United Kingdom | United States | |
|---|---|---|---|
| 2003 | 352,259 | 540,640 | 5,681,664 |
| 2004 | 343,121 | 540,409 | 5,790,761 |
| 2005 | 339,029 | 541,990 | 5,826,394 |
| 2006 | 327,797 | 542,045 | 5,737,615 |
| 2007 | 328,357 | 528,631 | 5,828,697 |
| 2008 | 323,657 | 522,247 | 5,656,839 |
| 2009 | 272,176 | 474,579 | 5,299,563 |
Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.
Scatter Plots
When both variables are quantitative, we can construct a scatter plot to organize the data. To construct a scatterplot , we use a horizontal axis for the observations of one variable and a vertical axis for the observations of the other. When picking which axis to use for each variable consider whether you suspect that one variable depends on the other. The independent variable will be on the x-axis and dependent will be on the y axis. Each pair of observations is then plotted as a point.
The summary of the ages and the mileages of a sample of 5 cars is shown below:
| Car | \(x\) (age in years) | \(y\) (mileage in miles) |
|---|---|---|
|
1 |
7 |
78524 |
|
2 |
2 |
12574 |
|
3 |
1 |
24914 |
|
4 |
5 |
65813 |
|
5 |
3 |
39824 |
Amelia plays basketball for her high school. She wants to improve to play at the college level. She notices that the number of points she scores in a game goes up in response to the number of hours she practices her jump shot each week. She records the following data:
| \(X\) (hours practicing jump shot) | \(Y\) (points scored in a game) |
|---|---|
| 5 | 15 |
| 7 | 22 |
| 9 | 28 |
| 10 | 31 |
| 11 | 33 |
| 12 | 36 |
Construct a scatter plot and state if what Amelia thinks appears to be true.
- Answer
-
Figure \(\PageIndex{2}\)
Yes, Amelia’s assumption appears to be correct. The number of points Amelia scores per game goes up when she practices her jump shot more.