Introduction to Statistics - Part 1Presentation Transcript
Quantitative Data Analysis: Statistics – Part 1
" ... while man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician. "
Overview
General Statistics
The Normal Distribution
Z-Tests
Confidence Intervals
T-Tests
~ THE GOLDEN RULE ~
Statistics NEVER replace
the judgment of the expert.
Approach to Statistical Research
Formulate a Hypothesis
State predictions of the hypothesis
Perform experiments or observations
Interpret experiments or observations
Evaluate results with respect to hypothesis
Refine hypothesis and start again
(Basically the same as all other research)
Hypothesis Testing
H 0 : Null Hypothesis , status quo
H A : Alternative Hypothesis , research question
So, either :
" The data does not support H 0 "
or
" We fail to reject H 0 "
Types of Data
Continuous
height, age, time
Discrete
# of days worked this week, # leaves on a tree
Ordinal
{Good, O.K., Bad}
Nominal
{Yes/No}, {Teacher/Chemist/Haberdasher}
Picturing The Data
Time-Series Plots
Time related Data
e.g. Stock Prices
Pie Charts
Nominal/Ordinal
Only suitable for data that adds up to 1
Hard to compare values in the chart
Bar Charts
Nominal/Ordinal
Easier to compare values than pie chart
Suitable for a wider range of data
Histograms
Continuous Data
Divide Data into ranges
Dot Plots
Nominal/Ordinal
Represents all the data
Difficult to read
Scatter Plots
Excellent for examining association between two variables
Box Plots
Nominal/Ordinal
1IQR, 3IQR - First interquartile range (IQR), third interquartile range (IQR)
Outliers
John Tukey
Born June 16, 1915
Died July 26, 2000
Born in New Bedford, Massachusetts
He introduced the box plot in his 1977 book," Exploratory Data Analysis "
Also the Cooley–Tukey FFT algorithm and jackknife estimation
While working with John von Neumann on early computer designs, Tukey introduced the word "bit" as a contraction of "binary digit". The term "bit" was first used in an article by Claude Shannon in 1948.
The term "software", which claims Paul Niquette coined in 1953, was first used in print by Tukey in a 1958 article in American Mathematical Monthly , and thus some attribute the term to him.
John Tukey Paul Niquette Claude Shannon John von Neumann
Question 1
In a telephone survey of 68 households, when asked do they have pets, the following were the responses :
16 : No Pets
28 : Dogs
32 : Cats
Draw the appropriate graphic to illustrate the results !!
Question 1 - Solution
Total number surveyed = 68
Number with no pets = 16
=>Total with pets = (68 - 16) = 52
But total 28 dogs + 32 cats = 60
=> So some people have both cats and dogs
Question 1 - Solution
How many? It must be (60 - 52) = 8 people
No pets = 16
Dogs = 20
Cats = 24
Both = 8
-------------------------
Total = 68
Question 1 - Solution
Graphic: Pie Chart or Bar Chart
Question 1 - Solution
Graphic: Pie Chart or Bar Chart
Pitfalls of Surveys
The Literary Digest Poll
1936 US Presidential Election
Alf Landon (R) vs. Franklin D. Roosevelt (D)
The Literary Digest Poll
Literary Digest had been conducting successful presidential election polls since 1916
They had correctly predicted the outcomes of the 1916, 1920, 1924, 1928, and 1932 elections by conducting polls.
These polls were a lucrative venture for the magazine: readers liked them; newspapers played them up; and each “ballot” included a subscription blank.
The Literary Digest Poll
In 1936 they sent out 10 million ballots to two groups of people:
prospective subscribers, “who were chiefly upper- and middle-income people”
a list designed to "correct for bias" from the first list, consisting of names selected from telephone books and motor vehicle registries
The Literary Digest Poll
Response rate: approximately 25%, or 2,376,523 responses
Result: Landon in a landslide (predicted 57% of the vote, Roosevelt predicted 40%)
The Literary Digest Poll
Response rate: approximately 25%, or 2,376,523 responses
Result: Landon in a landslide (predicted 57% of the vote, Roosevelt predicted 40%)
Election result: Roosevelt received approximately 60% of the vote
The Literary Digest Poll
POSSIBLE CAUSES OF ERROR
Selection Bias : By taking names and addresses from telephone directories, survey systematically excluded poor voters.
Republicans were markedly overrepresented
in 1936, Democrats did not have as many phones, not as likely to drive cars, and did not read the Literary Digest
“ Sampling Frame” is the actual population of individuals from which a sample is drawn: Selection bias results when sampling frame is not representative of the population of interest
The Literary Digest Poll
POSSIBLE CAUSES OF ERROR
Non-response Bias : Because only 20% of 10 million people returned surveys, non-respondents may have different preferences from respondents
Indeed, respondents favored Landon
Greater response rates reduce the odds of biased samples
Definitions and Formula
Terminology
Population: is a set of entities concerning which statistical inferences are to be drawn.
Sample: a number of independent observations from the same probability distribution
Parameter: the distribution of a random variable as belonging to a family of probability distributions, distinguished from each other by the values of a finite number of parameters
Bias: a factor that causes a statistical sample of a population to have some examples of the population less represented than others.
Outliers (and their treatment)
Outliers (and their treatment)
An "outlier" is an observation that does not fit the pattern in the rest of the data
Check the data
Check with the measurer
If reason to believe it is NOT real, change it if possible, otherwise leave it out (but note).
If reason to believe it is real, leave it out and note.
The Mean
The Mean (Arithmetic)
The mean is defined as the sum of all the elements, divided by the number of elements.
The statistical mean of a set of observations is the average of the measurements in a set of data
The Mode
The mode is defined as the most frequently element in a set of elements.
For example [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] has a mode of 6.
Given the list of data [1, 1, 2, 4, 4] the mode is not unique - the dataset may be said to be bimodal, while a set with more than two modes may be described as multimodal.
The Median
The median is defined as the middle element, or the value separating the higher half of a sample from the lower half.
If there is an even number of elements, it is half the sum of the middle two elements.
The Variance
But there can be a lot of variance in individual elements,
e.g. teacher salaries
Average = €22,000
Lowest = € 12,000
Difference = 12,000 - 22,000 = -10,000
The Variance
The Variance
The Variance
The Variance
Sum of (Sample - Average) = 0, thus we need to define variance.
The variance of a set of data is a cumulative measure of the squares of the difference of all the data values from the mean divided by sample size minus one.
Standard Deviation
The standard deviation of a set of data is the positive square root of the variance.
- 1 - 1
Born 27 March 1857
Died 27 April 1936
Born in Islington, London, England
Father of Mathematical Statistics
protégé of Francis Galton
Inventor of the P-value, the Pearson correlation coefficient, Chi distance, the Method of moments, and Principal Component Analysis
Karl Pearson
Karl Pearson the term "standard deviation" in 1893, "although the idea was by then nearly a century old" (Abbott; Stigler, page 328).
The term "standard deviation" was introduced in a lecture of 31 January 1893, as a convenient substitute for the cumbersome "root mean square error" and the older expressions "error of mean square" and "mean error."
The term was firist used in a publication in 1894 by Pearson in "Contributions to the Mathematical Theory of Evolution," (Philosophical Transactions of the Royal Society A, 185, (1894), 71-110.).
http://jeff560.tripod.com/s.html
Question 2
Find the mean and variance of the following sample values :