Box Plots

Statistical Measures

Measures of Central Tendency: numbers that represent the middle of the data (mean, median, mode)

Mean ( x ): Arithmetic average

Median: Middle of the data listed in ascending order(use if there is an outlier)

Mode: Most common number (if modes – can be more than one)

Statistics practice of analyzing a set of data

Skewed Right (positively):

Skewed Left (negatively):

Less data to the right.

Less data to the left

Standard Deviation (σ):

Variance: How much data is spread out

Measure of variation (high=spread out)

5 Number Summary:

Minimum Value (0 Percentile)

Q1: Quartile 1 (25th Percentile)

Med (Q2): Median (50th Percentile)

Min:

Q3: Quartile 3 (75th Percentile)

Max: Maximum or Q4 (100th Percentile)

Middle 50%

Q1 Q2

Med

Q3

Max

Q4

Min

IQR: Q3 – Q1 (where the middle 50% are)

25%

25%

25%

25%

Mean

Standard Deviation

Minimum1st QuartileMedian3rd QuartileMaximum

Calculator Information

You still have to find the mode by looking for the most common number(s), calculate the IQR by finding the diffference of Q3 and Q1

and finding the range by finding the difference of maximum and minimum

• Calculator Commands• Enter data into STATEDIT• STAT→CALC → 1-Var STATS…

Make the 130 a 120 and the 156 a 166. Recalculate What changed? Why?

Listed below are the weights of 10 people (in pounds)130, 150, 160, 145, 142, 143, 170, 132, 145, 156

Find the mean:

mode:

standard deviation

median:

minimum:

1st quartile:

3rd quartile:

maximum

IQR:

Range:

Make a box plot:

Skewed:

147.3

145

11.62

145

130

142

156

170

Q3-Q1: 156-142=14

Max-min: 170-130=40

Right (positive)

Standard deviation, Range

The data is more spread out

Class Data set of “The day of the month you were born on”

Find the mean:

mode:

standard deviation

median:

minimum:

1st quartile:

3rd quartile:

maximum

IQR:

Range:

Make a box plot:

Skewed:

The following is the amount of black M&M’s in a bag: 12, 13,14, 15, 15, 16, 17, 20, 21,22,23,24,25 Find the mean and standard deviation

The following is the amount of black M&M’s in a bag: 9, 10,11, 14, 15, 16, 17, 20, 21,23,26,27,28 Find the mean and standard deviation

Mean: 18.23Standard Deviation: 4.28

Mean: 18.23Standard Deviation: 4.28

Explain why the means are the same but the standard deviation is larger for the 2nd example.

The data is more spread out although it’s the same average.

Test Scores (n=60)

1. Between what scores do the middle 50% lie?2. Between what scores does the lowest 25% lie?3. Which range of scores has more density? (more numbers in a smaller number)4. Estimate how many people got between 85-89?5. Estimate how many people got below an 85?6. What is the IQR?7. What percentile did a person with a 70 get?

70-89

25%

25%

25%

25%

55-70

85-89

15

30

60*0.25 = 15

15 15 15 15

89-70 = 1925

60 70 80 90 100 110 120 130 140 145

Box plot of 80 Bowlers

1) What is the maximum score? 2) What is the IQR?3) What percentage of bowlers got above a 85?4) How many bowlers got below a 100?5) What percentile did a 120 get?6) Between what scores did the top 25% get?7) Where are there less density of bowlers?

25%

25%

25%

25%

80*.25=20

20 20 20 20

140120-85=35

25+25+25=75

20 + 20 = 40

75% (75% are below)

120-140

60-85

Which of the following will have the most variability?

A. Heights of people in this room

B. Ages of people in this room

C. The number of countries that people have been to in this room?

Variability: How close the numbers are togetherMore spread out= High Variability = Large Standard Deviation =High IQR

Which would have a lower standard deviation? (Be prepared to explain):

A. The heights of students in this class

B. The heights of students in this school

http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg

http://en.wikipedia.org/wiki/Skewness

Normal DistributionBell Curve

0-4 5-9 10-14 15-19 20-25 26-30 30 +0

1

2

3

4

5

6

Years of Teaching Experienc e at Forsyth High School

<5051-60

61-7071-80

81-90

91-100

101-110

111-120

121-130

131-140

141-150>150

05

101520

IQ's of Randomly Selected People

Number of Shoes Owned per Person

Frequency

0-5 1

6-10 6

11-15 10

16-20 11

21-25 9

>26 8

Determine if the following examples are Normally Distributed, Positively Skewed, or Negatively Skewed.

Place the following under negatively skewed, normally distributed, or positively skewed, or

random?A) The amount of chips in a bag

B) The sum of the digits of random 4-digit numbers?

C) The number of D1’s that students in this class have gotten?

D) The weekly allowance of students

E) Age of people on a cruise this week

F) The shoe sizes of females in this class

Debate:

• Side 1) You are trying to convince your teacher to always curve test grades to a standard deviation

• Side 2) You are trying to convince your teacher to never curve test grades to a standard deviation

The next few slides are done as a wrap up or warm-up the

next few days

There are three at the end for easy questions

Mode: Most often number.Mean: Average. Median: The middle number when arranged from smallest to largest.Best to show when there are outliers!!!

1) Find the mode, mean, and median: 5,7,9,9,30

2) Which is the largest?

3) Now include a 90 in the data. Which of the three changed the most?

4) When they list salaries, why do they state the median price and not the mean price?

9 12 9

Mean

Mean: It went from 12 to 25

Median is less affected by outliers

Deeper Understanding• Suppose there are 20 tests and the scores

are all an 80%. What would change if 2 more tests were added that were both a 90%, mean or median?

• What if there were 20 tests, 4 were 70%, 12 were 80%, and 4 were 90%. Three more tests were added to group scoring 70%, 90%, and 100%. How would the mean or median change?

Trick or Treat• Ten neighborhood kids went out to get candy.

Here is a list of the number of treats they received:

45, 34, 56, 32, 10, 32, 62, 11, 55, 34

a. Find the mean, median, and IQR of the treats.

b. The kid who got 62 treats, went back out and got 262 treats. Find the new mean, median and IQR.

c. Which does a better job of describing the typical number of treats for the new data? Why?

d. Draw a box plot.

Ex. Find the mean, median, and mode of 85, 76, 88, 91, 85, 58, 88, 91, 97, 91, 88, 97, 97

• Mean ( ) = median = , mode = x

Draw a box plot of the data.

FIVE-NUMBER SUMMARY:Find the 5 number summary and draw a

box plot.Maria: 8, 9, 6, 7, 9, 8, 8, 6, 9, 9, 8, 7, 8, 7, 9, 9, 7, 7, 8, 9

Min.:

Q1:

Q2 (median):

Q3:

Max.:

Interquartile Range (IQR): Q3 – Q1=

6

8 7

9

9

9 – 7 = 2

9 87 6

FIVE-NUMBER SUMMARY:

Gia: 8, 9, 9, 9, 6, 9, 8, 6, 8, 6, 8, 8, 8, 6, 6, 6, 3, 8, 8, 9

Min.:

Q1:

Q2 (median):

Q3:

Max.:

Interquartile Range (IQR): Q3 – Q1=

3

8

6

8.5 9

8.5 – 6 = 2.5

Find the 5 number summary and draw a box plot.

3 8 6

8.5

9