Download Chapter 3 Review MDM 4U

3.1-3.4 Review
Will be assessed on the Ch3
Summative
Mr. Lieff
3.1 Graphical Displays
 Name and be able to analyze the various types
of distributions


Symmetric: Uniform, U-shaped, Mound-shaped
Asymmetric: Left/Right-skewed
 How do you calculate bin width?

(range) ÷ (# of bars)
3.2 Central Tendency
 Calculate mean, median, mode and weighted mean
 Determine which measure is appropriate
Symmetric  Mean/Median
 No outliers  Mean
 Outliers  Median
 Qualitative data; frequency important  Mode
 Skewed distributions
 Right skewed: mode < median < mean
 Left-skewed: mean < median < mode

3.3 Measures of Spread
 Calculate and interpret Range, IQR and
Standard Deviation (4-6 data points)




A larger value for ANY measure of spread
means the data has more spread (less
consistent)
Range  size of the interval containing all of
the data
IQR  size of the interval containing the
middle 50%
Std. dev.  average deviation from the mean
3.3 Measures of Spread cont’d
 How to calculate IQR
OMLUD*
Order the data!!!
* = credit to Chris/Jasmine/Holly
 Find the median, Q2
 Find the 1st half median, Q1
 Find the 2nd half median, Q3
 IQR = Q3 – Q1
 How to calculate Std.dev.
 Find the mean
 Find the deviations (data point – mean)
 Square the deviations
 Average the deviations  variance σ2
 Take square root  std. dev. σ

3.4 Normal Distribution
 Know the characteristics of a Normal
Distribution (68–95–99.7% Rule)
 Calculate the % of data in an interval based
on std.dev.
 Ex: If a set of data has mean 10 and standard
deviation 2, what percent of the data lie
between 6 and 14?
 ans: 6 is 2 std dev below the mean and 14 is
2 std dev above. So 95% of the data falls in
the range (see next slide)
Normal Distribution
99.7%
95%
68%
X ~ N (10,2 2 )
34%
34%
0.15%
13.5%
13.5%
2.35%
2.35%
4
0.15%
6
8
10
12
14
16
Normal Distribution
 Ex: If a set of data has mean 10 and standard
deviation 2, what percent of the data lie
between 8 and 14?
 Ans: 34% + 34% + 13.5% = 81.5%
Chapter 3/5 Review
Normal Distribution and
Binomial Probabilities
Mr. Lieff
3.5 Z-Scores
2
 Standard Normal Distribution X ~ N (0, 1 )
 mean 0, std dev 1
 z-scores map any data to this distribution
 1) Calculate a z-score
 # of std. dev. above/below the mean
z
xx

 2) Calculate the % of data below / above a value
(z-score table on p. 398-399)
 3) Calculate the percentile for a piece of data (round
z-score table percentage to a whole number)
 4) Calculate the percentage of data between 2 values
(find z-scores, look up %s below both, subtract
smaller from larger)
3.5 Z-Scores
 Ex: Given that X~N(10,22), what percent of
the data is between 7 and 11?
 Ans:




for 7: z = (7 – 10) ÷ 2 = -1.5  6.68%
for 11: z = (11 - 10) ÷ 2 = 0.5  69.15%
69.15 – 6.68 = 62.47
So 62.47% of the data lies between these two
values
5.3 Binomial Distributions
 recognize a binomial experiment situation



n identical trials
two possible outcomes
independent events (constant probability)
 calculate probabilities for these situations
n k
nk
P( X  k )    p  1  p 
k 
5.3 Binomial Distributions
 ex: A family decides to buy 5 dogs. If the
chances of picking a male and female are
equal, what is the probability of picking exactly
3 males?
 ans: using binomial probability distribution
formula:
 5  1 
P( X  3)    
 3  2 
3
1
 
2
5 3
 0.3125
5.3 Binomial Distributions
 Calculate the expected value for # of passes on 4




tests if you have a 60% chance of passing each time
Ans: E(X) = np = 4(0.60) = 2.4
So you are expected to pass 2.4 tests
Expected value is an average  continuous
Don’t round in these situations
5.4 Normal Approximation of
Binomial Distribution
 What about finding a range of probabilities


e.g., What is the probability of picking between 1 and 4 males?
e.g., P(1 ≤ X ≤ 4) = P(X=1) + P(X=2) + P(X=3) + P(X=4)
 If the number of trials is large enough, the
binomial distribution approximates a Normal
Distribution
5.4 Normal Approximation of
Binomial Distribution
 Verify that a binomial distribution can be
approximated by a Normal distribution


np > 5
n(1 – p) > 5
 Calculate 𝑥 and σ, given the number of trials,
n, and the probability of success, p
 Use z-scores to calculate the probability of a
range of data (below or above a value, or
between two values)

Use boundary values ending in .5!
5.4 Normal Approximation of
Binomial Distribution
 ex: A die is rolled 100 times. What is the
probability of getting fewer than 15 sixes?
 Ans:
1
5
Check : np  100   16.66 n(1  p )  100   83.33
6
6
1
 1  1 
x  100   16.66   100 1    3.72
6
 6  6 
14.5  16.66
z
 0.58  28.1%
3.72
 From the z-score table, the probability is
28.1%
5.4 Normal Approximation of
Binomial Distribution
 ex: what is the probability of getting between 15
and 20 sixes?
 ans:
1
 1  1 
x  100   16.66   100 1    3.72
6
 6  6 
14.5  16.66
for 15 : z 
 0.58  28.1%
3.72
20.5  16.66
for 20 : z 
 1.03  84.9%
3.72
p  84.9  28.1%  56.8%
3.6 Mathematical Indices
 Arbitrary numbers that provide a measure of
something
 Given a set of data:

Work with a given Index formula or create your own

Interpret the meaning of calculated results
 Moving averages – use for data that fluctuates over
time
 e.g., 3-day moving average is the average of
every consecutive 3-day period (no
wraparounds)
 Ex: BMI, Slugging Percentage, Moving Average
Review
 Read through the class slides
 p. 199 #6; p. 200 #3-7 (use a table for #6)
 pp. 324 – 325 #7, 9-12
 16 Multiple Choice
 8 Problems
 You will be provided with:


Formulas in Back of Book
z-score table on p. 398