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