Stats_lecture_3 (Statistics lecture on bell
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Transcript Stats_lecture_3 (Statistics lecture on bell
Statistics lecture 3
Bell-Shaped Curves
and Other Shapes
Goals for lecture 3
Realize many measurements in nature
follow a bell-shaped (“normal”) curve
Understand and learn to compute a
standardized score
Learn to find the proportion of the
population that falls into a given range
Memorize the Empirical Rule
Histogram
Bell-Shaped “Normal” Curve
Bell-Shaped “Normal” Curve
Bell-Shaped “Normal” Curve
Bell-Shaped “Normal” Curve
Remember?
Mean (average): Sum of the values
divided by the number of values
Standard deviation: A measure of how
spread out the values are. Think of it as
the “average distance” of all values
from the mean.
Some Characteristics
of a Normal Distribution
Symmetrical (not skewed)
One peak in the middle, at the mean
The wider the curve, the greater the
standard deviation
Area under the curve is 1 (or 100%)
mean
Why it looks like that
With many things in nature, most
individuals fall near the average. The
farther you move above or below the
average, the fewer individuals there are
with those extreme values.
Examples: Height, weight, IQ, pulse rate
Bell-shaped wear
Bell-shaped wear
Not all curves are “normal”
Normal Curve...
If you know these two things:
The Mean
The Standard Deviation
...
...Normal Curve
...you
The
into
The
The
can figure these things:
proportion of individuals who fall
any range of values
percentile of any given value
value of any given percentile
Percentiles
Your percentile for a particular measure
(like height or IQ) is the percentage of
the population that falls below you. In
one of my recent classes:
My height (183 cm): 89th percentile
My weight (
): 99th percentile
My age (62): 99th percentile
104 kg
Standardized Scores
A standardized score (also called the
z-score) is simply the number of
standard deviations a particular value is
either above or below the mean.
The standardized score is:
Positive if above the mean
Negative if below the mean
Standardized Score Examples
Class height: Mean 170 cm, StdDev: 10 cm.
What is the z-score of someone:
160 cm
180 cm
175 cm
150 cm
170 cm
145 cm
Calculate z-score
for a Particular Value
z-score = (Value - mean) / StdDev
185 cm : (185 – 170) / 10 = 15 / 10 = +1.5
165 cm: (165 - 170) / 10 = -5 / 10 = -0.5
180 cm: (180 - 170) / 10 = 10 / 10 = +1.0
What’s the Point?
With z-score or percentile, you can
compare unlike things.
For instance, I am heavier (99th pctile)
than I am tall (89th pctile).
With a z-score, you can look up the
percentile in a table or an online
calculator
The Empirical Rule
For any normal curve, approximately:
68% of values within one StdDev of the
mean
95% of values within two StdDevs of
the mean
99.7% of values within three StdDevs
of the mean
Empirical Rule
Empirical Rule
Empirical Rule
Outlier
A value that is more than three
standard deviations above or below the
mean.
Apply Empirical Rule
to Class Height
Class height: Mean 170 cm., StdDev 10 cm.
About 68% of class is between what heights?
160 cm and 180 inches (+/- 10 cm)
About 95% of class is between what heights?
150 inches and 190 inches (+/- 20 cm)
Data visualization goals
See different ways of graphically
displaying data.
Learn the features of a good statistical
picture.
Be able to identify common problems
with graphs and plots.
Learn to read graphs comprehensively.
Why do we turn data
into graphics?
Easier to understand
Easier to see the trends
A good graphic will convey the same
message you would get if you really
studied the data
“Graphics reveal data.”
-- Edward Tufte
Two kinds of variables
Categorical: Data that can be counted
in categories, such as gender or race
Measurement: Data that can be
recorded as a number and then put into
order, such as IQ, weight, cigarettes
smoked per day, etc.
Pictures of
Categorical Data
Three common types of graphics for
categorical data:
Pie charts
Bar graphs
Pictograms
Pie Charts
Women
37%
Men
63%
Good for showing one categorical
variable, like gender
Show the percentage that falls into
each category
Bar Graphs
Can show two or
more categorical
variables
simultaneously
(for example, height
and gender)
10
8
Students
12
F
6
M
4
2
0
48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82
Inches
number
Pictograms
A
grades
B
C
D
F
Height of pictures is used like bars
Pictograms
can be misleading
We tend to focus on the area, rather
than just the height
Pictograms
can be misleading
To be fair, you should keep the width of
pictograms the same
Pictures of
Measurement Data
Lots of ways to illustrate measurement
variables:
Stemplots and histograms (lecture 2)
Line graphs (also called fever charts)
Scatter plots
Others: Area, radar, doughnut, highlow-close, surface plots, maps, et al.
Stemplots
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Line Graph
(Fever Chart)
Scatter Plot
Good for displaying the relationship
between two measurement variables
Scatter Plot
350
Doig
pounds
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150
100
60
65
70
inches
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Scatter Plot
350
Doig
pounds
300
250
200
150
100
60
65
70
inches
75
80
Scatter Plot
height vs. weight
pounds
300
250
200
150
100
60
65
70
inches
75
80
Scatter Plot
height vs. weight
300
pounds
250
200
150
100
60
65
70
inches
75
80
Scatter Plot
height vs. weight
300
pounds
250
200
150
100
60
65
70
inches
75
80
Difficulties and Disasters
Most common problems:
No labeling on one or more axes
Not starting at zero
Changes in labeling on axes
Misleading units
Graphs based on poor information
Checklist for Statistical Pictures
1. Does the message clearly stand out?
2. Is the purpose or title evident?
3. Is a source given for the data?
4. Did the data come from a reliable,
believable source?
5. Is everything labeled clearly and
unambiguously?
Checklist for Statistical Pictures
6. Do the axes start at zero?
7. Do the axes maintain a constant scale?
8. Are there breaks in the numbers on the
axes that may be easy to miss?
9. Have financial numbers been adjusted for
inflation?
10. Is there extraneous information cluttering
the picture or misleading the eye?
Perguntas?