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There are three kinds of lies - lies, damned lies and statistics. ~Benjamin Disraeli, commonly misattributed to Mark Twain. APSTAT PART ONE Exploring and Understanding Data. What is Statistics?. Chapters 1-3. What is Stat?. Book Says: A way of reasoning Collection of tools and methods

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apstat part one exploring and understanding data

There are three kinds of lies - lies, damned lies and statistics. ~Benjamin Disraeli, commonly misattributed to Mark Twain

APSTAT PART ONEExploring and Understanding Data

what is stat
What is Stat?
  • Book Says:
    • A way of reasoning
    • Collection of tools and methods
    • Helps us understand the world
    • Statistics is about variation
stat basics
Stat Basics
  • Individuals
    • Object described by a set of data
    • People (#1), cars, animals, groups…
  • Variables
    • Categorical (Qualitative)– Usually involves words
      • Examples: sex, advisor, social security #...
    • Quantitative – Involve #’s
      • Examples: age, height, income, test score…
displaying categorical data1
Displaying Categorical Data
  • Realtive Frequency tables:
    • Just roll up the %’s
displaying categorical data2
Displaying Categorical Data
  • Contingency Table
    • Two Way table

Age at first “Real Kiss” (ahhhhhhhhhhhh…)

marginal distribution
Marginal Distribution

Age at first “Real Kiss” (ahhhhhhhhhhhh…)

  • Conditional Distribution:
    • % of males whose first kiss came when they were 10-14
    • % of 20-24 year old first kissers who were male
the rest of chapters 1 3
The Rest of Chapters 1-3
  • Displaying the data
    • Pie Charts
    • Bar Charts
    • Blah Blah Blah….
  • Simpson’s Paradox – AP MC
  • Being Skeptical – Important for real life
    • 5 W’s + 1H
      • Ex: 4 out of 5 dentists….
    • Displaying data
      • Lies, Dammed Lies, and Statistics
histograms
Histograms
  • Remember bar graphs? Same, but different.
  • Think of sorting boxes…
    • Same size boxes
  • ON TI-83
    • Enter Data into L1 (STAT>EDIT)
    • Go to STAT PLOT (2ND Y=)
    • Change Options
    • Go to ZOOM Choose Stat OR Go to WINDOW

Change Options

Go to GRAPH

histograms1
Histograms
  • Make a histogram of the following data:
  • Age of Teachers At WPS

25, 34, 37, 42, 51, 43, 49, 35, 37, 65,

outliers
Outliers
  • An observation that is outside the pattern
    • For example, ages in this classroom

16, 17, 16, 17, 18, 17, 17, 16, 18, 36

  • Formula to determine (l8r, sk8r)
    • For now “potential” or “possible” outlier
describing a distribution
Center

Mean - Average

Median - Middle

Shape

Symmetric

Skewed

Uniform

Bell Shaped

Bi- or Multi-modal

Spread

Standard Deviation

Range

IQR

Weird-ness

Outliers

Gaps

Describing a distribution
stemplots
Stemplots
  • Basic
  • Split Stems
  • Back-To-Back
basic stemplot
Basic Stemplot
  • Boys Weight in class (pounds)

10

11

12

13

14

15

16

17

18

3 4 6 9 9

0 2 5 7 8 8

0 0 1 3 4 4 5 8 9

1

9

KEY: 10 8 = 108 pounds

split stem stemplot
Split Stem Stemplot
  • Boys Weight in class (pounds)

3 4

6 9 9

0 2

5 7 8 8

0 0 1 3 4 4

5 8 9

1

9

14

14

15

15

16

16

17

17

18

KEY: 10 8 = 108 pounds

back to back stemplot
Back to Back Stemplot
  • Girls vs. Boys Weight in class (pounds)

10

11

12

13

14

15

16

17

18

8

9 3

8 7 7 3

9 4 0

2

1

3 4 6 9 9

0 2 5 7 8 8

0 0 1 3 4 4 5 8 9

1

9

KEY: 10 8 or 8 10 = 108 pounds

slide19
Mean
  • Average! Add ‘em up and divide by n
  • Sample Mean denoted as x (x-bar)
  • Not Resistant to extreme measures
    • ie. Ages in Mrs. Smith’s Kindergarten Class
    • 4,5,4,4,4,5,5,4,4,4,5,5,4,4,5,39
median
Median
  • Middle! Line ‘em up (in order) and find the middle. If two share it, find their mean.
  • Resistant to extreme measures
    • ie. Ages in Mrs. Smith’s Kindergarten Class
    • 4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,39
quartiles
Quartiles
  • Median cuts data in half, Quartiles cut the Halves in Half!

Recall Teacher Ages:

25, 34, 35, 37, 37, 42, 43, 49, 51, 65

Median

3rd Quartile

Q3

1st Quartile

Q1

5 number summary
5-Number Summary
  • Low-Q1-Median-Q3-High
  • Shows Spread of Data

Recall Teacher Ages:

25, 34, 35, 37, 37, 42, 43, 49, 51, 65

  • 5-Number Summary:

25 35 39.5 49 65

boxplot
Boxplot
  • Graphical Representation of 5-Number Summary
  • Shows Shape, Spread, and Center
  • Always draw to scale:

25 35 39.5 49 65

outliers1
Outliers
  • First off, IQR – InterQuartile Range
    • Distance between Quartiles…

Recall Teacher Ages:

25, 34, 35, 37, 37, 42, 43, 49, 51, 65

  • IQR is 49-35=14
  • Outlier is anything 1.5 times IQR below Q1 or above Q3
  • Sooo…. An outlier would have to be 21 below 35 or 21 above 49…Below 14 or above 70. Nothing in our data is an outlier!
boxplot using ti 83
Boxplot Using TI-83

Enter Teacher Ages into L1 (clear old stuff first):

25, 34, 35, 37, 37, 42, 43, 49, 51, 65

  • ON TI-83
    • Go to STAT PLOT (2ND Y=)
    • Change Options
    • Go to ZOOM Choose Stat OR Go to WINDOW

Change Options

Go to GRAPH

variance standard deviation
Variance & Standard Deviation
  • Variance - s2
    • Average of Squared distances from mean
    • In example 26/5 = 5.2
  • Standard Deviation – s
    • Square Root of Variance
    • In example, about 2.28
  • Standard Deviation
    • Measure of Spread
    • Use with Mean
    • Non-Resistant
  • On TI-83 Now…..

STAT>CALC-1VARSTAT

Mean = 6

it s normal to deviate

It’s Normal to Deviate

Chapter 6 – The Normal Model

density curve

Mean, Median and Mode

Density Curve
  • Area under a density curve is always 1
  • Symmetric density curve:
density curve continued

Mean

Mode

Mean

Skewed to the Left

(tail trails to the left)

Skewed to the Right

(tail trails to the right)

Median

Density Curve Continued
  • Density curves are often skewed
  • Recall Median is “resistant” while Mean is not
histograms2

50% of Population

50% of Population

Histograms
  • Median is “equal areas” point
  • Mean is “balance point” – “think Physics”
normal distributions bell shaped

Concave

Down

Concave

Up

Concave

Up



+

Normal Distributions (bell shaped)
  • Center is mean m –(population mean)
  • Spread is Standard Deviation s – (population standard deviation)
    • To find, look for inflection points
68 95 99 7 rule

Raw-Score (X)

 2

 3

 1

 + 1

 + 2

 + 3

z-Score (z)

3

2

1

0

1

2

3

68 – 95 – 99.7 Rule
  • Also called EMPIRICAL RULE

Probability = 99.7% within 3

Probability = 95% within 2

Probability = 68% within 1

percentiles and quartiles
Percentiles (and quartiles)
  • Think standardized tests or class rankings
  • Percent of observations to the LEFT of an observation
  • Quartiles:
    • First is at 25th percentile
    • Median is at 50th percentile
    • Third is at 75th percentile
z score

Raw-Score (X)

 2

 3

 1

 + 1

 + 2

 + 3

z-Score (z)

3

2

1

0

1

2

3

Z-SCORE
  • Number of Standard Deviations (s) away from the Mean (m)
z score continued
Z-SCORE Continued
  • Example, You have an IQ of 148 The IQ test you took has a distribution N(105, 20). What is your Z-Score? What does this mean?
  • = population mean

 = population standard deviation,

X = Raw-Score,

z = z-Score

  • Normal Distribution Notation N (, )
using tables
Using Tables
  • Ex. – Your IQ Z-SCORE was 2.15. What does it mean now?
using tables1
Using Tables
  • Ex. – If someone’s IQ was at the 10th percentile, what would their Z-SCORE be?
using ti 83
Using TI-83
  • Normalcdf (Xlower, Xupper, , ) : - use to convert Raw-Score directly to probability.
  • Normalcdf (Zlower, Zupper) : - use to convert z-Score to probability

***For Graphics use Shadenorm (GTANG notes)

using ti 831
Using TI-83
  • Test Empirical Rule (68-95-99.7)
    • Find Normalcdf(-1,1), Normalcdf(-2,2), Normalcdf(-3,3)
  • Ex. What percent of IQ Scores would fall between 100 and 110 Using N(105, 20)? What percent would be above 150?
    • Normalcdf(100,110,105,20)
    • Normalcdf(150,1000000000,105,20)
normality
Normality
  • Just check Box and Whisker plot or Histogram on TI-83
  • ALWAYS do this if raw data is given
    • Sketch result and comment on it!