Five+Number+Summary,+Box+Plots

**__5 Number Summary Box__**
The five number summary box is a way to easily display a large set of data with fewer numbers. It reveals properties about the data without looking at the entire set.

__ **The 5 Numbers of the Box Plot:** __

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The 5 Number summary can be used to summarize a set of data. It contains Five Numbers :======

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1. **The Minimum**- the lowest point of the list of data =====

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2. **The Lower (First) Quartile** (Range and Interquartile range)- the middle term of the lower half of the data =====
 * *The lower half is determined by going 1 number to the left of the median of the entire data.
 * It can be though of as 1/4 of the data, or the median of the lower half.

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3. **The Median**- the middle value of the data set =====

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4. **The Upper (Second) Quartile**- the middle term or median of the upper half. =====
 * *The upper half is determined by going 1 number to the right of the median of the entire data.
 * It can be though of as 1/4 of the data, or the median of the upper half.

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5. **The Maximum-** the highest point of the list of data. =====

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__//Method One// __//: //=====

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2. After Arranging the data in the correct order, split it in two at the median, excluding the median, minimum and maximum =====

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3. The Median of the lower half is the Lower or First Quartile, and the Median of the upper half is the Upper or Second Quartile =====

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2. After Arranging the data, split data in two halves and use the median in both =====

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3. The Lower quartiles the median of the lower half, and the Upper Quartile is the median of the upper half =====

__** Non-SAT style questions: **__ //(Solutions at bottom of page)//

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__Example One__: Perform the problem using both method one and method two. =====

**Data: 3, 17, 29, 31, 45, 57, 68, 81, 96**

 * __Method One __:
 * **Sample Minimum**: 3
 * **Lower Quartile**: 23
 * **Median**: 45
 * **Upper Quartile**: 74.5
 * **Maximum**: 96


 * ===== __<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">Method Two __<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">: =====
 * =====**Sample Minimum**: 3=====
 * =====**Lower Quartile**: 29=====
 * =====**Median**: 45=====
 * =====**Upper Quartile**: 68=====
 * =====**Maximum**: 96=====

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__**<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">For more information on finding the parts of the 5 box summary: **__=====
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">[|Step by Step look]
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">[|Example and explanation]

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__**<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">A View of the Box Plot **__**<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">: **===== //<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;"> *An outlier is data that is numerically distant from the rest of the data* //
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;"> A Box Plot is a convenient way to graphically show your 5 number summary.
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">It may also show which observations are outliers

**__Skewness of Box Plots:__**

 * =====Left Skew: When a box plot is skewed to the left, as shown below, the mean value is less than the median. This is because there are more lower values than smaller ones.=====
 * EX: //shown below//


 * Right skew: When a box plot is skewed to the right, the mean value is greater than the median. This is because there are more larger values than smaller ones.
 * EX: //shown below//

For more information, check out: [|http://pirate.shu.edu/~wachsmut/Teaching/MATH1101/Descriptives/box.html]

**HOW TO MAKE A BOX PLOT**


 * 1) Draw an Y OR X-axis that encompasses all the statistics found in the five number summaries.
 * 2) Draw a box with ends at quartiles 1 and 3.
 * 3) Then you should draw a vertical line in the box to represent the median at its corresponding value along the axis.
 * 4) Next multiply 1.5 times the difference of quartile 2 and quartile 1 and call it x.
 * If the minimum value is greater than the distance between the first quartile and x, draw a whisker (line) to the minimum.
 * If the minimum is less than x, draw a whisker to x and mark any data values less than it with asterisks.
 * If the maximum value is less than the value of the third quartile plus x, draw a whisker to the maximum value.
 * If the max is greater than thew a whisker to x and mark any data values that are bigger than it with asterisks.




 * **//This would be an example of a box plot. The 75% value would be the upper quartile and the 25% value would be the lower quartile.//**
 * [|For more information on creating a box plot.]

Practice Problems:

__** Practice Problem One: **__
 * =====<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">Data: 108, 109, 119, 120, 134, 149, 154, 169, 183, 197 =====
 * =====**//Solutions to practice problem One://**=====
 * __Method One__:
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Sample Minimum**: 108
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Lower Quartile**: 119
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Median**: 141.5
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Upper Quartile**: 169
 * <span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Maximum**: 197


 * ===== __Method Two__: =====
 * =====<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Sample Minimum**: 108 =====
 * =====<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Lower Quartile**: 119.5 =====
 * =====<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Median**: 141.5 =====
 * =====<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Upper Quartile**: 161.5 =====
 * =====<span style="background-color: transparent; font-family: Arial; font-size: medium; text-decoration: none; vertical-align: baseline;">**Maximum**: 197 =====

Another sample question

The boxplots shown above summarize two data sets, I and II. Based on the boxplots, which of the following statements about these two data sets CANNOT be justified?

(A) The range of data set I is equal to the range of data set II. (B) The interquartile range of data set I is equal to the interquartile range of data set II. (C) The median of data set I is less than the median of data set II. (D) Data set I and data set II have the same number of data points. (E) About 75% of the values in data set II are greater than or equal to about 50% of the values in data set I.

__** Bibliography- **__

> [|%2Fapmathteacher.wikispaces.com%2Ffile%2Fview%2F2002%2BAP%2BStatistics%2BMultiple%2BChoice%2BExam.doc&ei=muGnT-XUN-H30gG_ks3GBQ&usg=AFQjCNFWtDJdSXsPGRMLGBfDqxQ42Z1nCg&sig2=oJbcH6K2D20Hdf_KhxTt_g]
 * //__http://faculty.valenciacollege.edu/slenhof/Statistical%20Methods/Notes/How%20to%20Find%20the%20Five%20Number%20Summary.pdf__//
 * __http://illuminations.nctm.org/Lessons...mSum.htm__
 * __http://www.statisticshowto.com/article...tistics/__
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