What Statistics Are Needed to Draw a Box Plot?

Learning Outcomes

• Display data graphically and interpret graphs: stemplots, histograms, and box plots.
• Recognize, describe, and summate the measures of location of data: quartiles and percentiles.

Box plots (also called box-and-whisker plots or box-whisker plots) requite a good graphical image of the concentration of the data. They also show how far the extreme values are from most of the data. A box plot is constructed from five values: the minimum value, the first quartile, the median, the third quartile, and the maximum value. We use these values to compare how shut other data values are to them.

To construct a box plot, apply a horizontal or vertical number line and a rectangular box. The smallest and largest information values characterization the endpoints of the centrality. The start quartile marks one end of the box and the tertiary quartile marks the other terminate of the box. Approximatelythe centre [latex]50[/latex] percent of the data autumn inside the box. The "whiskers" extend from the ends of the box to the smallest and largest information values. The median or 2d quartile can be between the get-go and third quartiles, or it can be 1, or the other, or both. The box plot gives a good, quick picture of the data.

Notation

You may encounter box-and-whisker plots that accept dots mark outlier values. In those cases, the whiskers are not extending to the minimum and maximum values.

Consider, again, this dataset.

[latex]1[/latex], [latex]1[/latex], [latex]ii[/latex], [latex]2[/latex], [latex]iv[/latex], [latex]six[/latex], [latex]6.8[/latex], [latex]7.2[/latex], [latex]8[/latex], [latex]8.three[/latex], [latex]9[/latex], [latex]10[/latex], [latex]x[/latex], [latex]eleven.five[/latex]

The first quartile is ii, the median is seven, and the tertiary quartile is 9. The smallest value is one, and the largest value is [latex]11.five[/latex]. The post-obit prototype shows the constructed box plot.

Note

See the reckoner instructions on the TI web site.

The two whiskers extend from the beginning quartile to the smallest value and from the 3rd quartile to the largest value. The median is shown with a dashed line.

Note

It is important to showtime a box plot with ascaled number line. Otherwise the box plot may not be useful.

Example

The following data are the heights of [latex]40[/latex] students in a statistics class.

[latex]59[/latex]; [latex]sixty[/latex]; [latex]61[/latex]; [latex]62[/latex]; [latex]62[/latex]; [latex]63[/latex]; [latex]63[/latex]; [latex]64[/latex]; [latex]64[/latex]; [latex]64[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]68[/latex]; [latex]68[/latex]; [latex]69[/latex]; [latex]70[/latex]; [latex]70[/latex]; [latex]70[/latex]; [latex]70[/latex]; [latex]70[/latex]; [latex]71[/latex]; [latex]71[/latex]; [latex]72[/latex]; [latex]72[/latex]; [latex]73[/latex]; [latex]74[/latex]; [latex]74[/latex]; [latex]75[/latex]; [latex]77[/latex]

Construct a box plot with the following backdrop; the calculator instructions for the minimum and maximum values too as the quartiles follow the example.

• Minimum value = [latex]59[/latex]
• Maximum value = [latex]77[/latex]
• Q1: Kickoff quartile = [latex]64.5[/latex]
• Q2: Second quartile or median= [latex]66[/latex]
• Q3: Third quartile = [latex]seventy[/latex]

1. Each quarter has approximately [latex]25[/latex]% of the data.
2. The spreads of the 4 quarters are [latex]64.5 – 59 = 5.5[/latex] (get-go quarter), [latex]66 – 64.5 = one.5[/latex] (second quarter), [latex]70 – 66 = four[/latex] (third quarter), and [latex]77 – 70 = 7[/latex] (fourth quarter). So, the 2d quarter has the smallest spread and the fourth quarter has the largest spread.
3. Range = maximum value – the minimum value = 77 – 59 = 18
4. Interquartile Range: [latex]IQR[/latex] = [latex]Q_3[/latex] – [latex]Q_1[/latex] = [latex]70 – 64.5 = 5.5[/latex].
5. The interval [latex]59–65[/latex] has more than [latex]25[/latex]% of the information and so it has more data in it than the interval [latex]66[/latex] through [latex]70[/latex] which has [latex]25[/latex]% of the data.
6. The middle [latex]50[/latex]% (middle half) of the data has a range of [latex]5.5[/latex] inches.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

To find the minimum, maximum, and quartiles:

Enter data into the list editor (Pres STAT 1:EDIT). If yous demand to articulate the list, arrow up to the name L1, printing CLEAR, then pointer down.

Put the data values into the list L1.

Press STAT and pointer to CALC. Press 1:1-VarStats. Enter L1.

Printing ENTER.

Utilise the down and up arrow keys to scroll.

Smallest value = [latex]59[/latex].

Largest value = [latex]77[/latex].

[latex]Q_1[/latex]: First quartile = [latex]64.v[/latex].

[latex]Q_2[/latex]: 2d quartile or median = [latex]66[/latex].

[latex]Q_3[/latex]: Tertiary quartile = [latex]lxx[/latex].

To construct the box plot:

Printing 4:Plotsoff. Printing ENTER.

Pointer down and then use the right pointer key to go to the fifth film, which is the box plot. Press ENTER.

Arrow downwardly to Xlist: Press 2nd 1 for L1

Arrow down to Freq: Press ALPHA. Press i.

Press Zoom. Press ix: ZoomStat.

Press TRACE, and use the arrow keys to examine the box plot.

Try Information technology

The following information are the number of pages in [latex]forty[/latex] books on a shelf. Construct a box plot using a graphing calculator, and state the interquartile range.

[latex]136[/latex]; [latex]140[/latex]; [latex]178[/latex]; [latex]190[/latex]; [latex]205[/latex]; [latex]215[/latex]; [latex]217[/latex]; [latex]218[/latex]; [latex]232[/latex]; [latex]234[/latex]; [latex]240[/latex]; [latex]255[/latex]; [latex]270[/latex]; [latex]275[/latex]; [latex]290[/latex]; [latex]301[/latex]; [latex]303[/latex]; [latex]315[/latex]; [latex]317[/latex]; [latex]318[/latex]; [latex]326[/latex]; [latex]333[/latex]; [latex]343[/latex]; [latex]349[/latex]; [latex]360[/latex]; [latex]369[/latex]; [latex]377[/latex]; [latex]388[/latex]; [latex]391[/latex]; [latex]392[/latex]; [latex]398[/latex]; [latex]400[/latex]; [latex]402[/latex]; [latex]405[/latex]; [latex]408[/latex]; [latex]422[/latex]; [latex]429[/latex]; [latex]450[/latex]; [latex]475[/latex]; [latex]512[/latex]

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This video explains what descriptive statistics are needed to create a box and whisker plot.

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For some sets of data, some of the largest value, smallest value, first quartile, median, and third quartile may exist the same. For instance, y'all might take a information set in which the median and the third quartile are the aforementioned. In this case, the diagram would non accept a dotted line inside the box displaying the median. The right side of the box would display both the third quartile and the median. For example, if the smallest value and the first quartile were both i, the median and the 3rd quartile were both five, and the largest value was seven, the box plot would look like:

In this case, at least [latex]25[/latex]% of the values are equal to one. 20-five percent of the values are between ane and five, inclusive. At least [latex]25[/latex]% of the values are equal to 5. The elevation [latex]25[/latex]% of the values fall between five and vii, inclusive.

Example

Examination scores for a college statistics form held during the day are:

[latex]99[/latex]; [latex]56[/latex]; [latex]78[/latex]; [latex]55.5[/latex]; [latex]32[/latex]; [latex]90[/latex]; [latex]eighty[/latex]; [latex]81[/latex]; [latex]56[/latex]; [latex]59[/latex]; [latex]45[/latex]; [latex]77[/latex]; [latex]84.5[/latex]; [latex]84[/latex]; [latex]lxx[/latex]; [latex]72[/latex]; [latex]68[/latex]; [latex]32[/latex]; [latex]79[/latex]; [latex]90[/latex]

Test scores for a college statistics class held during the evening are:

[latex]98[/latex]; [latex]78[/latex]; [latex]68[/latex]; [latex]83[/latex]; [latex]81[/latex]; [latex]89[/latex]; [latex]88[/latex]; [latex]76[/latex]; [latex]65[/latex]; [latex]45[/latex]; [latex]98[/latex]; [latex]90[/latex]; [latex]80[/latex]; [latex]84.5[/latex]; [latex]85[/latex]; [latex]79[/latex]; [latex]78[/latex]; [latex]98[/latex]; [latex]90[/latex]; [latex]79[/latex]; [latex]81[/latex]; [latex]25.5[/latex]

1. Detect the smallest and largest values, the median, and the start and tertiary quartile for the day class.
2. Find the smallest and largest values, the median, and the first and third quartile for the nighttime class.
3. For each data prepare, what percentage of the data is between the smallest value and the showtime quartile? the first quartile and the median? the median and the tertiary quartile? the tertiary quartile and the largest value? What percent of the information is between the first quartile and the largest value?
4. Create a box plot for each ready of data. Use one number line for both box plots.
5. Which box plot has the widest spread for the middle [latex]50[/latex]% of the data (the data between the first and third quartiles)? What does this mean for that set of data in comparison to the other set up of information?

Try It

The following data prepare shows the heights in inches for the boys in a form of [latex]40[/latex] students.

[latex]66[/latex]; [latex]66[/latex]; [latex]67[/latex]; [latex]67[/latex]; [latex]68[/latex]; [latex]68[/latex]; [latex]68[/latex]; [latex]68[/latex]; [latex]68[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]70[/latex]; [latex]71[/latex]; [latex]72[/latex]; [latex]72[/latex]; [latex]72[/latex]; [latex]73[/latex]; [latex]73[/latex]; [latex]74[/latex]

The following data set up shows the heights in inches for the girls in a form of [latex]40[/latex] students.

[latex]61[/latex]; [latex]61[/latex]; [latex]62[/latex]; [latex]62[/latex]; [latex]63[/latex]; [latex]63[/latex]; [latex]63[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]65[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]66[/latex]; [latex]67[/latex]; [latex]68[/latex]; [latex]68[/latex]; [latex]68[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]69[/latex]

Construct a box plot using a graphing calculator for each data set, and land which box plot has the wider spread for the middle [latex]fifty[/latex]% of the information.

instance

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

[latex]10[/latex]; [latex]10[/latex]; [latex]ten[/latex]; [latex]15[/latex]; [latex]35[/latex]; [latex]75[/latex]; [latex]90[/latex]; [latex]95[/latex]; [latex]100[/latex]; [latex]175[/latex]; [latex]420[/latex]; [latex]490[/latex]; [latex]515[/latex]; [latex]515[/latex]; [latex]790[/latex]

The five numbers used to create a box-and-whisker plot are:

• Min: [latex]10[/latex]
• [latex]Q_1[/latex]: [latex]15[/latex]
• Med: [latex]95[/latex]
• [latex]Q_3[/latex]: [latex]490[/latex]
• Max: [latex]790[/latex]

The following graph shows the box-and-whisker plot.

Try It

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

[latex]0[/latex]; [latex]5[/latex]; [latex]5[/latex]; [latex]15[/latex]; [latex]xxx[/latex]; [latex]xxx[/latex]; [latex]45[/latex]; [latex]50[/latex]; [latex]50[/latex]; [latex]threescore[/latex]; [latex]75[/latex]; [latex]110[/latex]; [latex]140[/latex]; [latex]240[/latex]; [latex]330[/latex]

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Concept Review

Box plots are a type of graph that can help visually organize data. To graph a box plot the following information points must be calculated: the minimum value, the starting time quartile, the median, the tertiary quartile, and the maximum value. Once the box plot is graphed, you can brandish and compare distributions of data.

References

Data from West Magazine.

Additional Resources

Apply the online imathAS box plot tool to create box and whisker plots.

reedaredy1976.blogspot.com

Source: https://courses.lumenlearning.com/introstats1/chapter/box-plots/

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