1. Looking at Data-Distributions
1.1-Displaying Distributions with Graphs

2. Basic definitions Data-numbers with a context
Eg. Your friends new baby weighed 10.5 pounds, we know that baby is quite large. But if it is 10.5ounces or 10.5kg, we know that it is impossible-the context makes the number informative
Individuals-objects described in the data(people,animals,things)
Variable-any property/characteristics of an individual(IQ scores of persons)
Distribution-of a variable tells us what values & how often(frequency of a variable)

3. Types of variables
categorical variable-places an individual into one of several categories(male/female, smoker/nonsmoker)
quantitative variable-takes numerical values for which arithmetic operations such as adding & averaging can be performed(shoe size,age)

4. How to represent data?
Categorical variables-can use Pie-chart & bar graphs
Eg. make a pie chart/bar graph for distribution of gender
Quantitative variables-can use histogram

5. Example 1-The color of your car(distribution of the most popular colors for 2005 model luxury cars made in North America
Color
Percent
Silver 20
White, pearl 18
Black 16
Blue 13
Light brown 10
Red 7
Yellow,gold 6
What percent of vehicles are some other color?
Make a bar graph?
Can we make a pie chart for the given colors?
Would it be correct to make a pie chart if you added an “Other” category?

6. Example 2-The density of the earth (the variable recorded was the density of the earth as multiple of the density of water)

7. Using TI-84 create a histogram
Discuss the shape, center, spread and outliers

8. Example3-Do women study more than men
Example3-Do women study more than men? Variable-minutes studied on a typical weeknight of a first-year college class
Here are the responses of random samples of 30 women and 30 men from the class:
Women 180,120,150,200,120,90,120,180,120,150,60,240,180,120,180,180,120, 180,360,240,180,150,180,115,240,170,150,180,180,120
Men
90,90,150,240,30,0,120,45,120,60,230,200,30,30,60, 120, 120, 120, 90,120,240,60,95,120,200,75,300,30,150,180
Examine the data. Why are you not surprised that more responses are multiples of 10minutes? We eliminated one student who claimed to study 30,000 minutes per night. Are there any other responses you consider suspicious?
Make a back-to-back stem plot to compare the two samples. That is, use one set of stems with two sets of leaves, one to the right and one to the left of the stems.(Draw a line on either side of the stems to separate stems and leaves.) Order both sets of leaves from smallest at the stem to largest away from the stem. Report the approximate midpoints of both graphs. Does it appear that women study more than men(or at least claim that they do)?

9. Answers
a) Most people round their answers. The students who claimed 0 minutes, 360 minutes and 300 minutes.
B)The stemplots suggest that women(claim to) study more than men. The approximate centers are 175 minutes for women and 120 minutes for men.

10. Looking at Data-Distributions
1.2-Describing Distributions with numbers

11. Mean =sum of numbers/ number of numbers
Mean & Median
Mean =sum of numbers/ number of numbers
Median=Middle value(when the numbers are in ascending order)
Example 1: 103,105,109,140,170 (Median is 109-the number in the (n+1)/2th position from the bottom of the list-n is number of values)
Example 2: 18,19,20,20,26,28(Median is 20- the avg of n/2 position number & n/2+1 position). Mean =21.83
Example 3: replace 28 in example 2 by 100 & re -compute mean and median?
18,19,20,20,26,100
Mean =33.83
Median-does not change
Mean is affected by outliers
Median is not affected by outliers
A measure of center alone can be misleading
Solution-need a measure of spread(variability)

12. Measuring spread
Quartiles
Example 4–Age of 10 students
26,19,20,18,20,19,19,19,19,21
Sort them in ascending order
18,19,19,19,19,19,20,20,21,26
Median =19 (Q2 )
First quartile=median of the lower half of data(Q1 )=19
Third quartile=median of the upper half of data(Q3 )=20

13. IQR(Inter quartile range)= Q3 - Q1
Five-number summary
Min Q1 Q2 Q3 Max
Box plot- Picture of the five number summary. Can be used to compare two distributions
IQR(Inter quartile range)= Q3 - Q1
Max Q3
IQR
Median(Q2 ) Q1
Min

14. The 1.5 X IQR rule for suspected outliers
Example 5(travel times to work in New York-in minutes)
10,30,5,25,40,20,10,15,30,20,15,20,85,15,65,15,60,60,40,45
(single peaked/right skewed/no center observation,but there is a center pair)
The five number summary
IQR= =27.5
Apply 1.5XIQR rule
Step 1:calculate 1.5 X IQR=1.5 x 27.5
Step 2: Calculate Q1 -(1.5 X IQR)= =-26.25
Step 3: Calculate Q3 +(1.5 X IQR)= =83.75
Any values outside of (-26.25,83.75) are flagged as outliers
The suspected outlier in the data is 85

15. Standard deviation(s)
Used as a measure of spread when mean=center
Units of s=same as data units
s always positive
Higher s->more spread
s=0->no spread -> all observations equal
s affected by outliers
Example :1,1,2,5,3

16. Formulas for variance & standard deviation

17. Looking at Data-Distributions
1.3 –Density Curves and Normal Distributions

18. Definitions
Density Curve-Special type of histogram such that total area under the curve is 1
Typical histogram
Example for a Density Curve
Relative frequency
Bin limits
Characteristics of density curve
All y values positive
total area under curve=1
Curve approaches to zero for extreme left & right x values

19. Definitions Normal Distribution Formula
It can be shown that the probability density function for a normal random variable, X, with mean X and standard deviation X has the following form.
TI-84 calculator-> 1)STAT plot off 2) enter in Y1-use normalpdf(x,mean,standard deviation) 3)normalpdf( ) found in 2nd->DISTR

20. Definitions The 68-95-99.7 rule
Example-When Mean 0 & standard deviation is 1
Approximately 68% of the observations fall within one standard deviation of the mean
Approximately 95% of the observations fall within two standard deviation of the mean
Approximately 99.7% of the observations fall within three standard deviation of the mean

21. Definitions
tables-allows us to calculate the probabilities for a normal distribution
How to get numbers?
There are too many normals(one per possible mean/one per possible standard deviation)
>infinitely many
Need to standardize
Standardization of Normal Random Variables. If X is normally distributed, its standardization is
Equation:

22. Definitions
Standard normal(Z) : N(0,1) , mean 0 & Standard deviation 1
Now can calculate the fraction of my data set between any two limits