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POPULATION VS. SAMPLE
• Population: a collection of ALL
outcomes, responses, measurements
or counts that are of interest.
• Sample: a subset of a population
• What are some examples of
population?
Ex: BHS
• What are some examples of samples?
• Ex: Sample of BHS could be the Senior
Class
INTERPRETING HISTOGRAMS
•Look at OVERALL PATTERN
•Center
•Spread
•Shape
•Symmetric?
•Skewed Right (tail right)?
•Skewed Left (tail left)?
•Unimodal, bimodal, multimodal?
•Look at striking DEVIATIONS
•Called OUTLIERS (lies outside the overall pattern)
PANCAKES VS.
SKYSCRAPERS
• Histograms with too many intervals 
Pancakes

Histograms with too few intervals  Skyscrapers
HISTOGRAM ON
CALCULATOR
• Stat, 1 (to enter data)
• Stat plot (to choose histogram)
• Zoom 9 (to set up axis)
• Window (to modify widths of bars)
WHY USE A STEMPLOT?
• Easier to find the middle
• Shows the shape of the distribution
DETAILS:
If
data has too many digits, you can round off:
4.1385  4.14
5.2273  5.23
If data falls into too few stems, you can split them up, 0-4
and 5-9 so each stem appears twice.
Babe Ruth Home Runs becomes:
2
2
3
3
4
4
5
5
6
2
5
4
5
11
66679
44
9
0
key 7 2 = 72
Median is a resistant measure of center
• When are the mean and median the same?
• When are they different?
• Is the mean or median enough?
– Only gives information about the center.
– We want to know about spread and variability
• Range – includes outliers
– So, it is often best to look at the middle two
quartiles (the middle half of the data)
Quartiles, divides data into 4 equal parts:
1. Arrange data in order and locate the median (also Q2)
2. The first quartile, Q1, is the median of the first half of the data
3. The third quartile, Q3, is the median of the second half of the data
Example: 5, 7, 10, 14, 18
Q1 = 10
19, 25, 29, 31, 33
Q2 = 18.5 Q3 = 29
Note: If odd number of data, do not include the median in your Q1 and Q3
Babe Ruth Data:
22, 25, 34, 35, 41, 41, 46, 46, 46, 47, 49, 54, 54, 59, 60
|
|
Q1 = 35
Q2 = 46
|
Q3 = 54
Interquartile Range (IQR) = Q3 - Q1 = 54-35 = 19
One rule of thumb to identify
outliers is to compute 1.5 *
IQR.
If a value falls
–above Q3 + 1.5 * IQR or
–below Q1 - 1.5 * IQR,
then the value is an outlier.
For example, with Babe Ruth
the IQR = 19.
So what is an outlier?
1.5*19 = 28.5
54 + 28.5 = 82.5 and 35 – 28.5 = 6.5
So, there are no outliers
FIVE NUMBER SUMMARY
• A convenient way to describe the center and spread of a data set is
the five number summary.
• The five number summary is defined as:
Min (value), Q1, Median, Q3, Max (value)
Here is an example: A Swiss study looked at the #
of hysterectomies performed by 15 male
doctors:
20 25 25 27 28 31 33 34 36 37 44 50 59 85 86
Find the 5 number summary for this data set.
Min Q1 Med Q3 Max
20 27 34 50 86
Here is a box plot of this data. A very powerful
graph
Box Plot
Collection 1
20
30
40
50
60
MaleDocs
70
80
90
** Must have a scale below the box plot – make the scale
first, then plot the five number summary.
CALCULATOR
INSTRUCTIONS
• BOXPLOT
– Enter data into list
– Stat plot
– Use down arrows to select the box plot and press ENTER
– Press ZOOM
– Press down arrow to ZoomStat and press ENTER
• 5 NUMBER SUMMARY
– Press STAT
– Press the right arrow to display the choices for STAT CALC
– Press ENTER to choose 1-Var Stats
BOXPLOT WITH
OUTLIERS
Box Pl ot
Coll ection 1
10
20
30
40
50
Male Do cs
60
70
80
90
ANOTHER MEASURE OF SPREAD:
STANDARD DEVIATION
• A measure of spread that we have
discussed is the 5 number summary. We
use that when using the median as measure
of center.
• A measure of spread used when using the
mean as measure of center is called
standard deviation.
Measure of Center
Measure of Spread
Median
5 number summary
Mean
standard deviation
CALCULATION OF STANDARD DEVIATION
1. Find the mean.
2. Find the difference between each data item and the
mean.
Distance from the mean.
3. Square each difference and add them.
Gets rid of any negatives and makes the larger differences even larger.
4. Find the average (mean) of these squared
differences, but need to divide by n-1 rather than n.
Average squared distance from the mean.
5. Take the square root of this average.
Just average distance from the mean.
HERE IS HOW TO COMPUTE THE STANDARD DEVIATION.
DATA: SET 12, 13, 13, 27, 27, 28. WHAT IS THE MEAN? 20
xi
xi-mean
(xi-mean)2
12-20 = -8
64
12
49
13-20 = -7
13
13-20 = -7
49
13
49
27-20 = 7
27
27-20 = 7
49
28-20 = 8
64
27
Add all
the (xi-mean)2 and divide by n-1 and take the
28
square root. 64+49+49+49+49+64= 324/(6-1)=64.8.
(64.8)=8.05
• The standard deviation (s) is the square root of the
variance.
• The variance (s2) is the average of the squares of the
differences of each observation from the mean.
• Here is the formula for standard deviation:
Where xi = each data point
xbar = sample mean
n = number of values
WHY N-1?
The idea of variance is the average squares
of the deviations of observations from the
mean. So why do we average by dividing by
n-1 instead of n?


The
of the deviations, no squares, is always 0.
So once we know n-1 of the deviations the nth one
is known also.We are not averaging n unrelated
numbers.
SO…
• The numbers are related.
– Only n-1 of the squared deviations can vary
freely so we average by dividing by n-1. n-1 is
called the degrees of freedom.
• Ex: If you have 4 markers and 4 people to choose
their markers, how many of them will have
FREEDOM of choice?
Probability Histogram Cumulative Histogram
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
are needed to see this picture.
ALWAYS PLOT
DATA FIRST!