Box Plots

Box plots (also called box-and-whisker plots or box-whisker plots) provide a good graphical photo of the concentration of the data. They also show how much the too much values space from many of the data. A crate plot is built from five values: the minimum value, the first quartile, the median, the third quartile, and also the preferably value. We use these values to compare just how close various other data values are to them.

You are watching: To make a boxplot of a distribution, you must know

To construct a box plot, usage a horizontal or vertical number line and also a rectangular box. The smallest and also largest data worths label the endpoints the the axis. The first quartile point out one finish of the box and also the third quartile marks the other end of the box. Approximately the center 50 percent that the data loss inside the box. The “whiskers” extend from the ends of the box to the smallest and largest data values. The median or second quartile deserve to be in between the first and third quartiles, or it have the right to be one, or the other, or both. The box plot gives a good, quick picture of the data.

You might encounter box-and-whisker plots that have actually dots noting outlier values. In those cases, the whiskers space not extending to the minimum and also maximum values.

Consider, again, this dataset.

1 1 2 2 4 6 6.8 7.2 8 8.3 9 10 10 11.5

The an initial quartile is two, the median is seven, and the 3rd quartile is nine. The smallest value is one, and the biggest value is 11.5. The adhering to image shows the built box plot.

Each quarter has around 25% the the data.The spreads of the four quarters are 64.5 – 59 = 5.5 (first quarter), 66 – 64.5 = 1.5 (second quarter), 70 – 66 = 4 (third quarter), and also 77 – 70 = 7 (fourth quarter). So, the 2nd quarter has actually the the smallest spread and also the 4th quarter has the biggest spread.Range = maximum value – the minimum value = 77 – 59 = 18Interquartile Range: IQR = Q3 – Q1 = 70 – 64.5 = 5.5.The expression 59–65 has an ext than 25% of the data so that has an ext data in it 보다 the interval 66 v 70 which has actually 25% of the data.The middle 50% (middle half) that the data has a range of 5.5 inches.

To uncover the minimum, maximum, and quartiles:

Enter data right into the perform editor (Pres STAT 1:EDIT). If you should clear the list, arrow up to the name L1, press CLEAR, and then arrow down.

Put the data values into the list L1.

Press STAT and arrow to CALC. Push 1:1-VarStats. Go into L1.

Press ENTER.

Use the down and also up arrow keys to scroll.

Smallest worth = 59.

Largest worth = 77.

Q1: very first quartile = 64.5.

Q2: second quartile or mean = 66.

Q3: third quartile = 70.

To construct package plot:

Press 4:Plotsoff. Push ENTER.

Arrow down and also then usage the right arrow key to go to the fifth picture, i m sorry is package plot. Push ENTER.

Arrow under to Xlist: Press second 1 because that L1

Arrow down to Freq: push ALPHA. Press 1.

Press Zoom. Push 9: ZoomStat.

Press TRACE, and also use the arrowhead keys to research the box plot.

The complying with data room the number of pages in 40 publications on a shelf. Build a crate plot making use of a graphing calculator, and state the interquartile range.

For part sets of data, some of the biggest value, smallest value, an initial quartile, median, and third quartile may be the same. Because that instance, you could have a data collection in which the median and also the 3rd quartile are the same. In this case, the diagram would not have actually a dotted heat inside the box displaying the median. The right side of the box would display both the 3rd quartile and the median. For example, if the the smallest value and the an initial quartile to be both one, the median and the third quartile were both five, and also the largest value to be seven, package plot would look like:

The first data set has the broader spread because that the center 50% the the data. The IQR for the first data set is better than the IQR for the 2nd set. This way that there is an ext variability in the center 50% the the first data set.

The following data collection shows the heights in inches because that the guys in a class of 40 students.

66; 66; 67; 67; 68; 68; 68; 68; 68; 69; 69; 69; 70; 71; 72; 72; 72; 73; 73; 74 The following data collection shows the heights in inches because that the girl in a course of 40 students. 61; 61; 62; 62; 63; 63; 63; 65; 65; 65; 66; 66; 66; 67; 68; 68; 68; 69; 69; 69 construct a crate plot making use of a graphing calculator because that each data set, and also state which box plot has actually the broader spread for the middle 50% the the data.

Graph a box-and-whisker plot for the data values shown.

The 5 numbers offered to develop a box-and-whisker plot are:

Min: 10Q1: 15Med: 95Q3: 490Max: 790

The complying with graph reflects the box-and-whisker plot.

Follow the steps you provided to graph a box-and-whisker plot for the data worths shown.

### Chapter Review

Box plots are a type of graph that can assist visually organize data. Come graph a crate plot the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and also the preferably value. Once the box plot is graphed, you have the right to display and compare distributions of data.

Use the adhering to information to answer the next two exercises. Sixty-five randomly selected vehicle salespersons were asked the variety of cars they normally sell in one week. Fourteen world answered that they typically sell 3 cars; nineteen generally sell four cars; twelve typically sell five cars; nine generally sell six cars; eleven usually sell seven cars.

Looking in ~ your box plot, go it show up that the data are concentrated together, spread out evenly, or focused in some areas, yet not in others? How can you tell?

More than 25% that salespersons sell 4 cars in a common week. You have the right to see this concentration in the box plot because the very first quartile is same to the median. The peak 25% and the bottom 25% space spread out evenly; the whiskers have actually the same length.

In a survey of 20-year-olds in China, Germany, and also the joined States, people were inquiry the variety of foreign countries they had actually visited in your lifetime. The following box plots screen the results.

In complete sentences, define what the shape of each crate plot implies around the distribution of the data collected.Have an ext Americans or an ext Germans surveyed to be to over eight international countries?Compare the 3 box plots. What do they imply around the foreign travel of 20-year-old inhabitants of the three nations when compared to each other?
Think of an instance (in words) wherein the data could fit right into the above box plot. In 2–5 sentences, create down the example.What does it mean to have actually the an initial and second quartiles therefore close together, if the second to 3rd quartiles are far apart?
Answers will vary. Possible answer: State University carried out a survey to view how connected its students room in community service. Package plot mirrors the variety of community company hours logged through participants over the past year.Because the an initial and 2nd quartiles are close, the data in this quarter is an extremely similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out out. This is clear because the 2nd quartile is so much away from the third quartile.
In complete sentences, describe why each statement is false.Data 1 has more data values over two than Data 2 has over two.The data set cannot have the exact same mode.For Data 1, there are much more data values listed below four 보다 there are over four.For i beg your pardon group, Data 1 or Data 2, is the worth of “7” much more likely to be an outlier? explain why in complete sentences.

A survey was conducted of 130 purchasers of brand-new BMW 3 collection cars, 130 purchasers of new BMW 5 series cars, and 130 purchasers of brand-new BMW 7 series cars. In it, world were asked the period they were as soon as they purchased your car. The adhering to box plots display the results.

In complete sentences, define what the shape of each box plot implies around the circulation of the data gathered for that car series.Which group is most likely to have an outlier? describe how you established that.Compare the three box plots. What do they imply about the period of purchase a BMW native the series when compared to every other?Look at the BMW 5 series. I m sorry quarter has actually the smallest spread of data? What is the spread?Look at the BMW 5 series. Which quarter has actually the biggest spread the data? What is the spread?Look at the BMW 5 series. Estimate the interquartile selection (IQR).Look at the BMW 5 series. Are there an ext data in the term 31 come 38 or in the interval 45 come 55? just how do you know this?Look in ~ the BMW 5 series. I m sorry interval has actually the fewest data in it? exactly how do you recognize this?31–3538–4141–64
Each crate plot is spread out out an ext in the greater values. Every plot is skewed to the right, for this reason the periods of the peak 50% of buyers are much more variable than the ages of the reduced 50%.The BMW 3 series is most most likely to have actually an outlier. It has the longest whisker.Comparing the median ages, younger people tend to buy the BMW 3 series, if older world tend to buy the BMW 7 series. However, this is not a rule, because there is so much variability in every data set.The second quarter has actually the smallest spread. There appears to be only a three-year difference between the first quartile and also the median.The 3rd quarter has the largest spread. There seems to be about a 14-year difference between the median and the 3rd quartile.IQR ~ 17 yearsThere is not enough information to tell. Each interval lies in ~ a quarter, so we cannot tell specifically where the data in that quarter is concentrated.The interval native 31 to 35 years has the fewest data values. Twenty-five percent the the values fall in the interval 38 to 41, and 25% fall in between 41 and 64. Because 25% of values fall in between 31 and 38, we recognize that fewer than 25% fall in between 31 and 35.

Twenty-five randomly selected students to be asked the number of movies lock watched the previous week. The outcomes are as follows:

# the moviesFrequency
05
19
26
34
41

Construct a box plot the the data.

Santa Clara County, CA, has around 27,873 Japanese-Americans. Their eras are as follows:

Age GroupPercent of Community
0–1718.9
18–248.0
25–3422.8
35–4415.0
45–5413.1
55–6411.9
65+10.3
Construct a histogram of the Japanese-American neighborhood in Santa Clara County, CA. The bars will certainly not be the exact same width for this example. Why not? What affect does this have on the dependability of the graph?What percent of the community is under age 35?Which box plot most resembles the info above?
For graph, examine student’s solution. 49.7% of the community is under the age of 35.Based top top the details in the table, graph (a) most closely represents the data.

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### Glossary

Box plota graph that gives a quick snapshot of the middle 50% the the dataFirst Quartilethe value that is the typical of the that the lower fifty percent of the ordered data setFrequency Polygonlooks like a heat graph however uses intervals to screen ranges of huge amounts that dataIntervalalso dubbed a course interval; an interval to represent a selection of data and also is offered when displaying huge data setsPaired Data Settwo data set that have a one come one connection so that: both data sets room the very same size, andeach data suggest in one data set is suitable with specifically one allude from the various other set.Skewedused to explain data that is not symmetrical; as soon as the ideal side of a graph looks “chopped off” compared the left side, we say the is “skewed to the left.” when the left next of the graph watch “chopped off” compared to the best side, us say the data is “skewed to the right.” Alternatively: when the lower values the the data are much more spread out, we say the data are skewed come the left. When the higher values are much more spread out, the data room skewed come the right.