Transcript Standard Deviation

Math I: Unit 2 - Statistics
Measures of Central Tendency: numbers that represent the middle
Mean ( x ):
Median:
Mode:
Arithmetic average
Middle of the data listed in ascending order
(use if there is an outlier)
Most common number
(can be more than one number or no numbers)
Measures of Variation: Variance, Standard Deviation
Variance (σ2): How much data is spread out
Standard Deviation (σ): Measure of variation from mean
(Large = spread out, Small = close together)
Range:
Difference between the Maximum and Minimum
Quartiles:
Separates ascending data into 4 equally sized(25%)
groups based on the how many data values
Inner Quartile Range (IQR):
Difference between 3rd and 1st Quartiles (Middle 50% of data)
5 Number Summary:
Min: Minimum Value (0 Percentile)
th Percentile)
Quartile
1
(25
Q1:
Med (Q2): Median (50th Percentile)
Q3: Quartile 3 (75th Percentile)
th Percentile)
Maximum
or
Q4
(100
Max:
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)
Min
25%
25% Med
Q1
25%
Q2
25%
Q3
Max
Q4
IQR: Q3 – Q1
Range: Max – Min
Skewed Left (negatively):
Skewed Right (positively):
Less data to the left (spread out)
Less data to the right (spread out).
Calculator Commands: One Variable Statistics
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]
Mean
Standard
Deviation
Minimum
1st Quartile
Median
3rd Quartile
Maximum
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
Example #1: Listed below are the weights of 10 people (in lbs)
130, 150, 160, 145, 142, 143, 170, 132, 145, 156
130
147.3
Mean: __________________
Minimum: _________
145 (x2)
Mode: __________________
142
1st quartile: _________
11.62
Standard Deviation: __________________
145
Median: _________
14 = 156 – 142 = Q – Q1
IQR: __________________ 3
156
3rd quartile: _________
40 = 170 – 130 = Max – Min
Range: ________________
Maximum: _________
170
Skewed:
Positive(Right), Negative (Left), or Normal
Make a box plot for the weights:
#1b: Change the 130 to a 120
and the 156 to a 166. Recalculate
What changed?
Standard deviation, Range
Why?
The data is more spread out
Class Data set of “The day of the month you were born on”
Mean: __________________ Minimum: _________
Mode: __________________ 1st quartile: _________
Standard Deviation:
Median: _________
__________________
3rd quartile: _________
IQR: __________________
Maximum: _________
Range: ________________ Make a box plot for the days:
Skewed: Positive(Right),
Negative (Left), or Normal
PRACTICE: Find the mean and standard deviation
#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
Mean: 18.23
Standard Deviation: 4.28
#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: 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.
Interpreting Boxplots: Test Scores (n=60)
.25*60 = 15
1. How many test scores are in each quartile?
2. Between what scores do the middle 50% lie? 70-89
3. Between what scores does the lowest 25% lie? 55-70
3. Which range of scores has more density?
85-89
(more numbers in a smaller number)
4. Estimate how many people got between 85-89? 15
5. Estimate how many people got below an 85?
30
6. What is the IQR?
89-70 = 19
7. What percentile did a person with a 70 get?
25
Box plot of 80 Bowlers
60
70
80
90
100
110
120
130
140 145
1) Estimate the values of the five-number summary
140
85 Med = _____
60
100 Q3 = 120
Min = ____Q1
= _____
_____ Max = _____
2) What is the number of bowlers in each quartile? 80*.25=20
4) What is the IQR?
140
120 – 85 = 35
5) What percentage of bowlers got above a 85?
25 + 25 + 25 = 75
6) How many bowlers got below a 100?
20 + 20 = 40
7) What percentile did a 120 get?
75% (75% are below)
8) Between what scores did the top 25% get?
120 to 140
9) Where is the lowest density of bowlers?
First Quartile: 60 to 85
3) What is the maximum score?
VARIABILITY:
How close the numbers are together
MORE spread out data:
= High Variability
= Large Standard Deviation
= High IQR
LESS spread out data:
= 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
Normal Distribution: “Bell Curve”
“Equal amount of data” to left and right of middle
Skewed Left (Negatively):
Skewed Right: (Positively)
Less data (spread out) to the Left
Less data (spread out) to the Right
Determine if the following examples are
Normally Distributed, Positively, or Negatively Skewed.
Years of Teaching Experience
Negatively(Left)
Positively(Right)
6
5
4
3
2
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
1
0
0-4
5-9
10-14
15-19
20-25
26-30
30 +
Normally
IQ's of Randomly Selected People
>150
141-150
131-140
121-130
111-120
101-110
91-100
81-90
71-80
61-70
51-60
<50
20
15
10
5
0
Determine if the following examples are
Normally Distributed, Positively, or Negatively Skewed.
Negatively
(Left)
Positively(Right)
Normally
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
9
12
9
2) Which is the largest? Mean
3) Now include a 90 in the data. Which of the three
changed the most? Mean: It went from 12 to 25
4) When they list salaries, why do they state the median
price and not the mean price?
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, 34
a. Find the mean, median, and IQR of the treats.
a. The kid who got 62 treats, went back out and got 262
treats. Find the new mean, median and IQR.
a. Which does a better job of describing the typical
number of treats for the new data? Why?
b. 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: 6
Q1: 7
Q2 (median):
Gia: 8, 9, 9, 9, 6, 9, 8, 6, 8,
6, 8, 8, 8, 6, 6, 6, 3, 8, 8, 9
Min: 3
Q1: 6
8
Q2 (median): 8
Q3: 9
Max: 9
Interquartile Range (IQR):
9–7=2
Q3: 8.5
Max.: 9
Interquartile Range (IQR):
8.5 – 6 = 2.5
6
6
7
8
9
3
8.5
9
8