Statistical Reasoningfor everyday life

Intro to Probability and Statistics

Mr. Spering – Room 113

4.2 Shapes of Distribution CLASS WORK:

Worksheet REVIEW

ACTIONS are REMEMBERED, WORDS can

be FORGOTTEN!

MAKE an EFFORT, NOT an EXCUSE

4.2 Shapes of Distribution Variation:

Describes how widely data are spread out about the center of a distribution.

????How would you expect the variation to differ between the heights of NCAA Division 1A Men’s College Basketball Centers and the heights of all High School Boy Basketball Players????

NCAA Division 1A Centers less variation High School Boy Basketball Players more variation

4.3 Measures of Variation

How do we investigate variation? Study all of the raw data… Range… Quartiles… Five-number summary (BOXPLOT or BOX-and-WHISKER)… Interquartile range… Semi-quartile range… Percentiles… MAD… Variance & Standard Deviation…

4.3 Measures of Variation

RANGE: The range of a

distribution is the difference between the highest and lowest data values.

range highest lowest

4.3 Measures of Variation

Find the range of the data.

4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3, 11.0 Range = 11.0 – 4.1

= 6.9

range highest lowest

4.3 Measures of Variation Misleading range:

Which Quiz Set has greater variation?

Quiz Set 1:

1, 10, 10, 10, 10, 10, 10, 10, 10, 10

Quiz Set 2:

2, 3, 4, 5, 6, 7, 8, 9, 10, 8, 9, 10, 6, 5

** Even though Set 1 has a greater range than Set 2 (9 > 8). Set 2 has a greater variation because Set 1 contains an outlier. Therefore, we use quartiles.**

4.3 Measures of Variation Quartiles:

Quartiles divide the data into four quarters. Lower Quartile (1st Quartile): is the median of the data

values in the lower half of a data set. Exclude the middle value in the data set if the number of data points is odd.

Middle Quartile (2nd Quartile): is the overall median Upper Quartile (3rd Quartile): is the median of the data

values in the upper half of a data set. Exclude the middle value in the data set if the number of data points is odd.

4.3 Measures of Variation Find quartiles…Example 1 – Upper and lower quartiles

Data6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36

Ordered data 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49

Median (2nd Quartile) 41

Upper quartile (3rd Quartile) 43

Lower quartile (1st Quartile) 15

4.3 Measures of Variation Find quartiles…

Example 2 – Range and quartilesA year ago, Angela began working at a computer store. Her supervisor

asked her to keep a record of the number of sales she made each month.

The following data set is a list of her sales for the last 12 months:34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37

Use Angela's sales records to find:a) the medianb) the rangec) the upper and lower quartiles

4.3 Measures of Variation Answers The values in ascending order are:

1, 11, 15, 19, 20, 24, 28, 34, 37, 47, 50, 57.

a) Median  = (6th + 7th observations) ÷ 2            = (24 + 28) ÷ 2            = 26

b) Range = difference between the highest and lowest values              = 57 - 1              = 56

4.3 Measures of Variationc) Lower quartile = value of middle of first half of data Q1

                           = the median of 1, 11, 15, 19, 20, 24                           = (3rd + 4th observations) ÷ 2                           = (15 + 19) ÷ 2                           = 17

d) Upper quartile = value of middle of second half of data Q3                       = the median of 28, 34, 37, 47, 50, 57                       = (3rd + 4th observations) ÷ 2                       = (37 + 47) ÷ 2                       = 42

These results can be summarized as follows:

4.3 Measures of Variation

Five-number summary: Consists of the following…

1) Low Value

2) Q1 (lower quartile)

3) Q2 (median)

4) Q3 (upper quartile)

5) High Value

#

4.3 Measures of Variation BOXPLOT or BOX-and-

WHISKER: Box plots show variation along the

number line.Steps for creating a box plot:

1. Draw a number line that spans the entire data set.

2. Above the number line, enclose the values from the lower to the upper quartile in a box.

3. Draw a line through the box at the value corresponding to the median.

4. Add “whiskers” extending to the low and high values.

Vertical box plot showing “normal” distribution

“FORESHADOWING”

4.3 Measures of Variation Example of 5 number summary and box plot.

Lowest Value 1

First Quartile (Q1) 6.5

Median (Q2) 12

Third Quartile (Q3) 19.5

Highest Value 24

So for the data set 1, 4, 9, 12, 12, 16, 23, 24 here is our box plot:

                                                                            

4.3 Measures of Variation

Right-SkewedLeft-Skewed Symmetric

Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3

Digest of BOXPLOTS and SKEWNESS

4.3 Measures of Variation Below is a Box-and-Whisker plot for the

following data:

0 2 2 2 3 3 4 5 5 10 27

The data are right skewed, as the plot depicts

0 2 3 5 270 2 3 5 27

Min Q1 Q2 Q3 Max

4.3 Measures of Variation Interquartile range:

i.e. If the five number summary is low: 3, high: 23, Q1: 4, Q2: 12, Q3: 19.

Then the interquartile range is IQR: (Q3-Q1) = (19 – 4) =15.

Interquartile rangeThe interquartile range is another range used as a measure of the variation. The difference between upper and lower quartiles (Q3–Q1), which is called

the interquartile range, also indicates the dispersion of a data set. The inter-quartile range spans 50% of a data set, and eliminates the influence of outliers because, in effect, the highest and lowest quarters are removed.Interquartile range = upper quartile (Q3) minus lower quartile (Q1)

4.3 Measures of Variation Next Time:

Semi-quartile range… Percentiles… MAD… Variance & Standard Deviation…

According to the box-n-whisker above what are the values for the 5 number summary:Low: 12Q1: 22Q2: 31Q3: 45High: 50

4.3 Measures of Variation Classwork:

PRACTICE MAKES PERMANENT

Pg 174 # 2-6 even and # 25-27 (Letters a, b only)