Math I: Unit 2 - StatisticsMeasures of Central Tendency: numbers that represent the middle

Arithmetic average

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

Mode: Most common number (can be more than one number or no numbers)

Standard Deviation (σ):

Variance (σ2): How much data is spread out

Measure of variation from mean (Large = spread out, Small = close together)

Mean ( x ):

Measures of Variation: Variance, Standard Deviation

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)

Range: Difference between the Maximum and Minimum

Inner Quartile Range (IQR):Difference between 3rd and 1st Quartiles (Middle 50% of data)

Quartiles: Separates ascending data into 4 equally sized(25%) groups based on the how many data values

Q1 Q2

Med

Q3

Max

Q4

Min 25% 25% 25%25%

IQR: Q3 – Q1

Range: Max – Min

Boxplot: “Box and Whisker” Whiskers represent outside quartiles (Min to Q1 and Q3 to Max)

Boxes Represent inside quartiles (Q2 to Med and Med to Q3)

Skewed Right (positively): Skewed Left (negatively): Less data to the right (spread out). Less data to the left (spread out)

Minimum1st QuartileMedian3rd QuartileMaximum

Mean

Standard Deviation

Input Data: [STAT] [EDIT] L1 DO NOT DELETE Lists: Highlight L1 [Clear] to start new list of data

Get Statistics from Data: [STAT] [CALC] [1: 1-Var STATS] [ENTER]

Calculator Commands: One Variable Statistics

REQUIRED Statistics by Hand!• Identify the MODE by looking for the most common number(s)

Use Five-Number Summary to calculate• IQR with Q3 and Q1

• RANGE with maximum and minimum

#1b: Change the 130 to a 120 and the 156 to a 166. Recalculate What changed? Why?

Example #1: Listed below are the weights of 10 people (in lbs)

130, 150, 160, 145, 142, 143, 170, 132, 145, 156

Standard deviation, Range

The data is more spread out

Mean: __________________

Mode: __________________

Standard Deviation: __________________

IQR: __________________

Range: ________________

Skewed: Positive(Right), Negative (Left), or Normal

Minimum: _________

1st quartile: _________

Median: _________

3rd quartile: _________

Maximum: _________

Make a box plot for the weights:

130142145156170

14 = 156 – 142 = Q3 – Q1

40 = 170 – 130 = Max – Min

147.3145 (x2)

11.62

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

Mean: __________________

Mode: __________________

Standard Deviation:

__________________

IQR: __________________

Range: ________________Skewed: Positive(Right), Negative (Left), or Normal

Minimum: _________

1st quartile: _________

Median: _________

3rd quartile: _________

Maximum: _________Make a box plot for the days:

#1: 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

#2: 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

Mean: 18.23 Standard Deviation: 4.28

Mean: 18.23 Standard Deviation: 6.24

#3: 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.

PRACTICE: Find the mean and standard deviation

Interpreting Boxplots: Test Scores (n=60)

1. How many test scores are in each quartile?2. Between what scores do the middle 50% lie?3. 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?

70-8955-70

85-89

1530

89-70 = 1925

.25*60 = 15

60 70 80 90 100 110 120 130 140 145

Box plot of 80 Bowlers

1) Estimate the values of the five-number summaryMin = ____Q1 = _____ Med = _____ Q3 = _____ Max = _____

2) What is the number of bowlers in each quartile?

3) What is the maximum score?

4) What is the IQR?

5) What percentage of bowlers got above a 85?

6) How many bowlers got below a 100?

7) What percentile did a 120 get?

8) Between what scores did the top 25% get?

9) Where is the lowest density of bowlers?

80*.25=20140120 – 85 = 3525 + 25 + 25 = 75

20 + 20 = 40

75% (75% are below)120 to 140First Quartile: 60 to 85

60 85 100 120 140

VARIABILITY: How close the numbers are together

MORE spread out data:

LESS spread out data:

= High Variability= Large Standard Deviation= High IQR

= Low Variability= Small Standard Deviation= Low IQR

#1) 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?

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

A.Heights of students in this class

B.Heights of students in this school

Skewed Right: (Positively) Skewed Left (Negatively): Less data (spread out) to the RightLess data (spread out) to the Left

Normal Distribution: “Bell Curve”“Equal amount of data” to left and right of middle

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

1

2

3

4

5

6

Years of Teaching Experience

<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(# of people)

0-5 1

6-10 6

11-15 10

16-20 11

21-25 9

>26 8

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

Positively(Right)

Normally

Negatively(Left)

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

Positively(Right)

Normally

Negatively(Left)

Positively(Right)

Normally Negatively (Left)

DEBATE:Think about possible PROS and CONS of each

• 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

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

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?

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 9Mean

Mean: It went from 12 to 25

Median is less affected by outliers

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, 34a. 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.

PRACTICE 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):

6

8 7

9 9

9 – 7 = 2

9 87 6

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.:

3

8 6

8.5 9

8.5 – 6 = 2.5

Interquartile Range (IQR):

3 8

6 8.5 9