STATISTICS Summarizing, Visualizing and Understanding Data.

STATISTICS

Summarizing, Visualizing and Understanding Data

I. Populations, Variables, and Data

Populations and Samples

To a statistician, the population is

the set or collection under investigation. Individual members of the population are not usually of interest. Rather, investigators try to infer with some degree of confidence the general features of the population.

Examples

Students currently enrolled at a certain university.

Registered voters in a certain Congressional district.

The population of large-mouthed bass in a certain lake.

The population of all decay times of a radioactive isotope.

Statistical Inference

Drawing and quantifying the reliability of conclusions about a population from observations on a smaller subset of the population.

Sample: The subset observed.

Variables and Data

A population variable is a descriptive number or label associated with each member of a population.

The values of a population variable are the various numbers (or labels) that occur as we consider all the members of the population.

Values of variables that have been recorded for a population or a sample from a population constitute data.

Types of Data

Nominal variables are variables whose values are labels.

Ordinal variables are variables whose values have a natural order.

Interval variables have values represented by numbers referring to a scale of measurement.

Ratio variables have values that are positive numbers on a scale with a unit of measurement and a natural zero point.

Guess the Type

Age Questionnaire responses: 1=”strongly

agree”,2=”agree”…,5=”strongly disagree”

Letter grades Reading comprehension scores Gender Zip codes Molecular velocities

II. Summarizing Data

Location Measures (Measures of Central Tendency)

A location measure or measure of central tendency for a variable is a single value or number that is taken as representing all the values of the variable. Different location measures are appropriate for different types of data.

The Mean

For interval or ratio variables x N individuals in the sample or population xi = value of x for ith individual

)(1

21 NxxxN

x

The mean of a population variable is denotedby (the Greek letter mu).

The Mean with Repeated Values

Distinct values of x: nj = frequency of occurrence of

Mxxx ,,, 21 jx

)(1

2211 MM xnxnxnN

x

The Mean with Repeated Values

Relative frequencies: N

nf jj

MM xfxfxfx 2211

Example

-2 1 3 4 6

2 1 3 5 3

jx

jn

The Median

Informally, the “middle” value when all the values are arranged in order

A number m is a median of x if at least half the individuals i in the population have

and at least half of them havemxi

mxi

The Median – Example 1

x: –2.0, 1.5, 2.2, 3.1, 5.7 (no repetitions)

median(x)=2.2


x: -2.0, 1.5, 3.1, 3.1, 3.1

median(x) = 3.1


x: -2.0, 1.5, 3.1, 5.7, 5.9, 7.1 median(x)=Any number in [3.1,5.7] By convention, for an even number

of individuals choose the midpoint between the smallest and largest medians, e.g.,

.4.42

7.51.3

m

Example

Change 7.1 to 71. What happens to the mean and the median?

The mean changes from 3.55 to 14.2

No change in the median The median is much less sensitive

to outliers (which may be mistakes in recording data)

A A- B+ B B- C+ C C- D+ D D- F

8 5 10 18 18 15 14 6 4 1 1 0

The Median for Ordered Categories

N=100. The median grade is B-.

The Mode

The data value with the greatest frequency

Not useful for interval or ordinal data if recorded with precision

The only useful location measure for strictly nominal data

A A- B+ B B- C+ C C- D+ D D- F

8 5 10 18 18 15 14 6 4 1 1 0

Example

The modes are B and B-.

Cumulative Frequencies and Percentiles

x is an interval or ratio variable. Ordered distinct values:

Relative frequencies:

Mxxx 21

Mfff ,,, 21

Cumulative Frequencies and Percentiles

Cumulative Frequencies

Cumulative Relative Frequencies

MM nnnN

nnnN

nnN

nN

21

3213

212

11

MM fffF

fffF

ffF

fF

21

3213

212

11

The Weather Person’s Prediction Errors x

x'j -2 1 3 4 6

nj 2 1 3 5 3

Nj 2 3 6 11 14

fj .1429 .0714 .2143 .3571 .2143

Fj .1429 .2143 .4286 .7857 1.000

Exercise

From the table above, what fraction of the data is less than 1? What fraction is greater than 3? What fraction is greater than or equal to 3?

Percentiles

x: an interval or ratio variable A number a is a pth percentile of x if at

least p% of the values of x are less than or equal to a and at least (100-p) % of the values of x are greater than or equal to a.

The 25th percentile is called the first quartile of x and the 75th percentile is the third quartile of x.

The 50th percentile is the second quartile or median.

Example

For the weather person’s errors, the 25th percentile is 3. The 50th percentile and third quartile are both 4.

Measures of Variability

Statisticians are not only interested in describing the values of a variable by a single measure of location. They also want to describe how much the values of the variable are dispersed about that location.

Population Variance and Standard Deviation

x: an interval or ratio variable. N=number of individuals in

population. Variance of x:

Standard deviation of x:

N

xxx N22

22

12 )()()(

2

Sample Variance and Standard Deviation

n: the number of individuals in a sample from a population

Sample variance:

Sample standard deviation:

1

)()()( 222

212

n

xxxxxxs n

2ss

Alternative Formulas for the Variance

Using frequencies:

Using relative frequencies:

N

xnxnxn MM22

222

112 )()()(

2222

211

2 )()()( MM xfxfxf

The Interquartile Range

Q1, Q3 : 1st and 3rd quartiles, respectively

Interquartile range:

Not influenced by a few extremely large or small observations (outliers)

13 QQIQR

The Range

The difference between the largest data value and the smallest

Range of sample values is not a reliable indicator of the range of a population variable

III. Graphical Methods

Pie Charts (Circle Graphs)

Sources: AT&T (1961) The World’s TelephonesR: A language and environment for statistical computing, the R core

development team.

Bar Charts (Bar Graphs)

Pros and Cons

Bar chart has a scale of measurement – more precise information

Pie chart gives more vivid impression of relative proportions, e.g., obvious at a glance that N. America had more than half the telephones in the world.

Stemplots (Stem and Leaf Diagrams)

Stem|Leaves Cumulative Frequency 4 | 7 1 5 | 448889 7 6 | 34789 12 7 | 012234455666888889999 33 8 | 0022234457799 46 9 | 0457 50

Grades of 50 students on a test

Find the Median


25th and 26th leaves circled. Median = 78

Exercise


The 1st quartile is 70 and the 3rd quartile is 82.

Boxplots (Box and Whisker Diagrams)

Elements of a Boxplot

box

whisker

quartiles median

outlierlargest

Boxplot Shows Distribution Skewed to the Left

Histograms

For interval or ratio data

Data is grouped into class intervals

Superficially like a bar chart

Frequency Histogram

Source: R: A language and environment for statistical computing, the R core development team.

Class interval (bin)

Height=bin frequency

Probability Histogram

Area of bar = relative bin frequencyE.g., .011×25=.275

Ogives(Cumulative Frequency Polygons)

Related to probability histograms

Examples of cumulative distribution functions

Probability histograms are examples of density functions

Example Ogive

Relationship Between Probability Histogram and Ogive

The height of the ogive is the cumulative area under the histogram

Estimating Percentiles from Ogives

Horizontal line has height .75

Vertical line intersects horizontal axis at 60

Estimated 3rd quartile is 60

True 3rd quartile is 62

Scatterplots (Scatter Diagrams)

Used for jointly observed interval or ratio variables

Example: Heights and weights of individuals

Example: State per capita spending on secondary education and state crime rate

Example: Wind speed and ozone concentration

Example Scatterplot

centroid

Fitting a Line

Relationship between variables x and y is approximately linear.

Approximately, y = a + bx. Find a and b so that data comes

closest to satisfying the equation. Least squares – a formal

mathematical technique to be shown later.

Line Fitted by Least Squares

IV. Sampling

Why Sample?

Because the population is too large to observe all its members.

The population may be partly inaccessible.

The population may even be hypothetical.

Statistical Inference

Drawing conclusions about the population based on observations of a sample.

Reliability of inferences must be quantifiable.

Random sampling allows probability statements about the accuracy of inferences.

Sampling With Replacement

Population has N members. n population members chosen

sequentially. Once chosen, a member of the population

may be chosen again. At each stage, all members of the

population are equally likely to be chosen. Random experiment with possible

equally likely outcomes.

nN

Sampling With Replacement (continued)

x is a population variable.

X1 = value of x for 1st sampled individual, X2 = value of x for 2nd sampled individual, etc.

Each Xi is a random variable. The random variables are independent.

The sequence is a random sample of values of x, or a random sample from the distribution of x.

nXXX ,,, 21

nXXX ,,, 21

Sampling Without Replacement

Population has N individuals. n members chosen sequentially. Once chosen, an individual may not

be chosen again. At each stage, all of the remaining

members are equally likely to be chosen next.

Random experiment with possible equally likely outcomes.

)1()1( nNNN

Sampling Without Replacement (continued)

Sample without replacement. Ignore the order of the sequence of

individuals in the sample. Random experiment whose

outcomes are subsets of size n. Experiment has possible

equally likely outcomes. Common meaning of “random

sample of size n”

)!(!

!, nNn

NC nN

Random Number Generators

Calculators and spreadsheet programs can generate pseudorandom sequences.

Press the random number key of your calculator several times.

Simulates a random sample with replacement from the set of numbers between 0 and 1 (to high precision).

Generating a Sample with Replacement

Number the individuals from 1 to N. Generate a pseudorandom number

R. Include individual i in the sample if

Repeat n times. Individuals may be included more than once.

iNRi 1

Exercise

Suppose you have 30 students in your class. Use the procedure just described to obtain a sample of size 10 (a) with replacement, (b) without replacement.

V. Estimation

The Sample Mean and Standard Deviation

is a random sample from the distribution of a population variable x.

The sample mean is

The sample variance is

nXXX ,,, 21

)(1

21 nXXXn

X

])()()[(1

1 222

21

2 XXXXXXn

S n

The Sample Mean and Standard Deviation (continued)

The sample standard deviation is

The sample mean, variance and standard deviation are all random variables because they depend on the outcome of the random sampling experiment.

2SS

Estimators

The sample mean, variance, and standard deviation have distributions derived from the distribution of values of the population variable x.

They are estimators of the population mean , the population variance 2, and the population standard deviation of x.

Unbiased Estimators

The theoretical expected values of the sample mean and sample variance are equal to their population counterparts, i.e.,

and S2 are said to be unbiased estimators of and 2, respectively

S is biased.

)(XE 22 )( SEand

X

)(SE

The Distribution of the Random Variable

The mean of is , the same as the mean of the population variable x.

The standard deviation of is

These are the theoretical mean and standard deviation.

X

X

X ./ n

Density Functions

A density function is a nonnegative function such that the total area between the graph of the function and the horizontal axis is 1.

A probability histogram is a density function.

Other density functions are limits of histograms as the number of data elements grows without bound.

The Standard Normal Density Function

Percentiles of the Standard Normal Distribution

za is the 100(1-) percentile of the distribution

Symmetry About the Vertical Axis

Probabilities Related to the Standard Normal Distribution

Other Normal Distributions

Let Z be a random variable with the standard normal distribution.

The mean of Z is 0 and the standard deviation of Z is 1.

Let and be any numbers, >0. Let Y =Z+ Y has the normal distribution with

mean and standard deviation .

Other Normal Distributions Example

= 1 and = 1.5

Standardizing: The Inverse Operation

Let Y be normally distributed with mean and standard deviation .

Let . This is the z-score of Y.

Then Z has the standard normal distribution and

Y

Z

][][

bZ

aPbYaP

The Central Limit Theorem

Let be the sample average of a random sample of n values of a population variable x.

The population variable x has mean and standard deviation .

Standardize by subtracting its mean and dividing by its standard deviation

X

X

)(

/

Xn

n

XZ

The Central Limit Theorem (continued)

Get Ready for the Central Limit Theorem!

The Central Limit Theorem(continued)

The Central Limit Theorem: As the sample size n grows without

bound, the distribution of Z approaches the standard normal distribution. This is true no matter what the distribution of values of the population variable x.

Another Statement of the CLT

For sufficiently large sample sizes n and for all numbers a and b,

In almost all applications, n≥50 is large enough.

])()(

[][

bnZ

anPbXaP

The CLT in Action

Sample n=30 from the population variable COUNTS whose distribution is tabulated. Calculate the sample average. Repeat this 500 times and construct a histogram of the z-scores of the 500 sample averages. Note: The distribution of COUNTS is very far from normal.

xj 0 1 2 3 4 5 6

fj .36 .33 .19 .08 .02 .01 .01

Distribution of COUNTS

Result-500 Averages of 30 Samples from COUNTS

Estimating a Population Mean

The sample mean is an unbiased estimator of the population mean .

For “large” sample sizes n, has approximately a normal distribution with mean and standard deviation

For large n, the sample mean is an accurate estimator of the population mean with high probability.

X

X

n/

Example

Suppose and we want to estimate with an error no greater than 0.05.

Assume is exactly normally distributed. Standardize.

X

X

]025.0|[|]05.0|[| nZPXP

Probabilities of 1-place Accuracy = 2

Confidence Intervals for the Population Mean – Review of 2/z

100(1-)% Confidence Interval

By the CLT

Rearranging the inequalities

][1 2/2/n

zXn

zXP

]/

[1 2/2/ zn

XzP

A Difficulty

is probably unknown, so the confidence interval

can’t be used. What to do?

nzX

2/

Enhanced Central Limit Theorem

Define the modified z-score for as

As n grows without bound, the distribution of Z approaches the standard normal distribution.

X

S

Xn

nS

XZ

)(

/

A More Useful Confidence Interval

By the enhanced CLT

An approximate 100(1-)% confidence interval is

][1 2/2/n

SzX

n

SzXP

n

SzX 2/

Example

n=50 from COUNTS ( = 1.14) = 1.32 S = 1.39 1- = .95 = =1.32±0.39

95% confidence interval: (0.93, 1.71) Don’t say .95=P[0.93<<1.71]

X

n

SzX 2/

50

39.196.132.1

Confidence Intervals for Proportions

x is a population variable with only two values, 0 and 1.

Numerical code for two mutually exclusive categories, e.g., “male” and “female”, or “approves” and “disapproves”.

p=relative frequency of x=1. =p; 2=p(1-p)

Confidence Intervals for Proportions (continued)

Sample n values of x, with replacement. Result is a sequence of 1s and 0s.

Sample mean is the relative frequency in the sample of 1s, e.g., the relative frequency of females in the sample of individuals.

Denote the sample mean by since it is an estimator of p.

p̂

Confidence Intervals for Proportions(continued)

By the enhanced CLT, is

approximately standard normal.

An approximate 100(1-

confidence interval is

)ˆ1(ˆ

)ˆ(

pp

ppnZ

n

ppzp

)ˆ1(ˆˆ 2/

Example

A public opinion research organization polled 1000 randomly selected state residents. Of these, 413 said they would vote for a 1¢ sales tax increase dedicated to funding higher education. Find a 90% confidence interval for the proportion of all voters who would vote for such a proposal.

Solution

n = 1000

1-=.90;

= 0.413 ± 1.645

(0.387, 0.489)

413.01000

413ˆ p

645.105.2

zz

n

ppzp

)ˆ1(ˆˆ 2/

1000

587.0413.0

Linear Regression and Correlation

x and y are jointly observed numeric variables, i.e., defined for the same population or arising from the same experiment.

Have observations for n individuals or outcomes.

Data: )11 ,(,),,( nn yxyx

Examples

(An observational study) Let x be the height and y the weight of individuals from a human population.

(A designed experiment) Let x be the amount of fertilizer applied to a plot of cotton seedlings and let y be the weight of raw cotton harvested at maturity.

Data on Fertilizer and Cotton Yield

x 2 2 2 4 4 4 6 6 6 8 8 8

y 2.3 2.2 2.2 2.5 2.9 2.7 3.4 2.7 3.4 3.5 3.4 3.3

Scatterplot of Fertilizer vs. Yield

Assumptions of Linear Regression

There is a population or distribution of values of y

for any particular value of x.

There are unknown constants a and b so that for

any particular value of x, the mean of all the

corresponding values of y is

The standard deviation of the values of y

corresponding to a value of x is the same for all

values of x.

bxay

The Method of Least Squares

Estimate a and b by choosing them to minimize the sum of squared differences between the observed values yi and their putative expected values

In symbols, minimize ibxa

2222

211 )()()( nn bxaybxaybxay

The Least Squares Estimates

Let and be the means of the observed x’s

and y’s. Let be the sample variance of the x’s.

The covariance between the x’s and the y’s is

The least squares estimate of the slope is

The least squares estimate of the intercept is

x y

)])(())([(1

111 yyxxyyxx

ns nnxy

2x

xy

s

sb

2xs

xbya

Least Squares Line for Cotton Yield

Correlation

The correlation between the x’s and y’s is

r is related to the slope b of the least squares regression line by

r is always between -1 and 1. r measures how nearly linear the relationship between x and y is. If r = 0, then x and y are uncorrelated.

yx

xy

ss

sr

y

x

s

sbr

Examples

STATISTICS Summarizing, Visualizing and Understanding Data.

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