Sampling and Descriptive Statics

7/28/2019 Sampling and Descriptive Statics

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Sampling and Descriptive Statistics

Berlin Chen

Department of Computer Science & Information Engineering

National Taiwan Normal University

Reference:

1. W. Navidi. Statistics for Engineering and Scientists. Chapter 1 & Teaching Material


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Sampling (1/2)

Definition: A o ulation is the entire collection of ob ects

or outcomes about which information is sought

All NTNU students

Definition: A sample is a subset of a population,

containing the objects or outcomes that are actuallyobserved

. .,

Choose the 100 students from the rosters of football or

basketball teams (appropriate?) Choose the 100 students living a certain dorm or enrolled in

the statistics course (appropriate?)

Statistics-Berlin Chen 2


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Sampling (2/2)

Definition: A sim le random sam le SRS of size n is a

sample chosen by a method in which each collection ofn

population items is equally likely to comprise the sample,us as n e o ery

Definition: A sample of convenience is a sample that isno rawn y a we - e ne ran om me o

Things to consider with convenience samples:


Only use when it is not feasible to draw a random sample


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More on SRS (1/3)

Definition: A conce tual o ulation consists of all the

values that might possibly have been observed

It is in contrast to tangible () population E.g., a geologist weighs a rock several times on a sensitive scale.

Each time, the scale gives a slightly different reading

.

that the scale could in principle produce



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More on SRS (2/3)

A SRS is not uaranteed to reflect the o ulation

perfectly

SRSs always differ in some ways from each other,

occasionally a sample is substantially different from the

population

Two different samples from the same population will vary

from each other as well

This phenomenon is known as sampling variation



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More on SRS (3/3)

The items in a sam le are inde endent if knowin thevalues of some of the items does not help to predict the

values of the others

(A Rule of Thumb) Items in a simple random sample

encountered in practice The exception occurs when the population is finite and the

samp e compr ses a su s an a rac on more an o epopulation

However, it is possible to make a population behave asthough it were infinite large, by replacing each item after


Sampling With Replacement


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Other Sampling Methods

Weighting Sampling Some items are given a greater chance of being selected than

others

. .,others

Stratified Sampling

The population is divided up into subpopulations, called strata A simple random sample is drawn from each stratum

Su ervised ?

Cluster Sampling

Items are drawn from the population in groups or clusters E.g., the U.S. government agencies use cluster sampling to

sample the U.S. population to measure sociological factors suchas income and unemployment


Unsupervised (?)


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Types of Experiments

One-Sam le Ex eriment

There is only one population of interest

A single sample is drawn from it

Multi-Sample Experiment

ere are wo or more popu a ons o n eres

A simple is drawn from each population

-

comparisons among populations



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Types of Data

Numerical or uantitative if a numerical uantit is

assigned to each item in the sample

Height Weight

Age

Categorical or qualitative if the sample items are placed

Gender

Hair color

Blood type



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Summary Statistics

The summar statistics are sometimes called descri tive

statistics because they describe the data

Numerical Summaries Sample mean, median, trimmed mean, mode

Sample standard deviation (variance), range

,

Skewness, kurtosis

.

Graphical Summaries

Stem and leaf plot Dotplot

Histogram (more commonly used)

Scatterplot

. Statistics-Berlin Chen 10


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Numerical Summaries (1/4)

Definition: Sam le Mean (thecenterofthedata) Let be a sample. The sample mean isnXX ,,1

n1

Its customary to use a letter with a bar over it to denote a

in 1

Definition: Sample Variance (howspreadoutthedataare) Let be a sam le. The sam le variance isnXX ,,1

n

i XXs

22 1

Which is equivalent to

2

1

22

1

1XnX

ns

n

ii


5040,30,20,,10

3231,30,29,,20


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Actually, we are interest in

Population mean

Population deviation: Measuring the spread of the population

mean

, ,

use sample mean to replace it

a ema ca y, e ev a ons aroun e samp e mean

tend to be a bit smaller than the deviations around the

So when calculating sample variance, the quantity divided by

rather than provides the right correctionn1n


To be proved later on !


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Definition: Sam le Standard Deviation

Let be a sample. The sample deviation isnXX ,,1

ii XX

ns

11

nnXs 22

1

The sample deviation also measures the degree of spread in a

in 11

samp e av ng esameun sas e a a



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If is a sam le, and ,where andnXX ,,1 bXaY a b

are constants, then XbaY

If is a sample, and ,where and

are constants, thennXX ,,1 ii bXaY a b

xyxy sbssbs and222

Definition: Outliers

Sometimes a sample may contain a few points that are much

larger or smaller than the rest (mainlyresultingfromdataentryerrors)



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More on Numerical Summaries (1/2)

Definition: The median is another measure of center of a

sample , like the mean

To compute the median items in the sample have to be ordered

nXX ,,1

y e r va ues

If is odd, the sample median is the number in positionn 2/1n

If is even, the sample median is the average of the numbers inpositions andn

2/n 12/ n

The median is an important (robust) measure of center for

samples containing outliers



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More on Numerical Summaries (2/2)

Definition: The trimmed mean of one-dimensional data

is computed by

First, arranging the sample values in (ascending or descending)or er

Then, trimming an equal number of them from each end, say,

%

Finally, computing the sample mean of those remaining



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More on mean

Arithmetic mean n

XX1

Geometric mean

in 1

nn

1

Harmonic mean

ii

1

1n

1 i iXn1

meanharmonic:1mean;arithmetic:1

minimum:maximum;:

mm

mm

mn

i

m

iXn

X1

meanquadratic:2

meangeometric:0

m

m

Arithmetic mean Geometric mean Harmonic mean

n

Weighted arithmetic meanStatistics-Berlin Chen 17http://en.wikipedia.org/wiki/Mean

ni i

i ii

wX 1

1


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Quartiles

Definition: the quartiles of a sample divides it asnXX ,,1

near y as poss e n o quar ers. e samp e va ues ave

to be ordered from the smallest to the largest

, .

The second quartile found by computing the value 0.5(n+1)

The third quartile found by computing the value 0.75(n+1)

Example 1.14: Find the first and third quartiles of the

.30 75 79 80 80 105 126 138 149 179 179 191

223 232 232 236 240 242 254 247 254 274 384 470

n=24

To find the first quartile, compute (n+1)25=6.25

105+126 /2=115.5


To find the third quartile, compute (n+1)75=18.75

(242+245)/2=243.5


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Percentiles

Definition: Thepth percentile of a sample , for anXX ,,1

number between 0 and 100, divide the sample so that as

nearly as possiblep% of the sample values are less than

.

Order the sample values from smallest to largest

Then com ute the uantit /100 n+1 where n is the sam le

size If this quantity is an integer, the sample value in this position is

. ,

either side

, ,

is the 50th percentile, and the third quartile is the 75th



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Mode and Range

Mode The sample mode is the most frequently occurring values in a

sample

Ran e

The difference between the largest and smallest values in asample

values



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Numerical Summaries for Categorical Data

For cate orical data, each sam le item is assi ned a

category rather than a numerical value

Two Numerical Summaries for Categorical Data

Definition: (Relative) Frequencies

The frequency of a given category is simply the number ofsample items falling in that category

Definition: Sample Proportions (alsocalledrelativefrequency) The sample proportion is the frequency divided by the

sample size



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Sam le Statistics and Po ulation Parameters 1/2

A numerical summar of a sam le is called a statistic

A numerical summary of a population is called a

parameter If a population is finite, the methods used for calculating the

numerical summaries of a sample can be applied for calculating

outcome) occurs with probability? See Chapter 2) Exceptions are the variance and standard deviation (?)

2

2

2

2

1:Normal

x

exf

However, sample statistics are often used to estimate

parameters (to be taken as estimators)


In practice, the entire population is never observed, so the

population parameters cannot be calculated directly


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Sample Statistics and Population Parameters (2/2)

A Schematic Depiction

Population Sample

Statistics

In erence

Parameters



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Graphical Summaries

Recall that the mean, median and standard deviation,

etc., are numerical summaries of a sample of of a

population

On the other hand, the graphical summaries are used as

will to help visualize a list of numbers (or the sampleitems). Methods to be discussed include:

Dotplot

Histo ram more commonl used Boxplot (more commonly used)

Scatterplot



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Stem-and-leaf Plot (1/3)

A sim le wa to summarize a data set

Each item in the sample is divided into two parts

stem, consisting of the leftmost one or two digits leaf, consisting of the next significant digit

The stem-and-leaf plot is a compact way to represent the

data It also gives us some indication of the shape of our data



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Example: Duration of dormant () periods of thee ser Old Faithful in Minutes

e s oo a e rs ne o e s em-an - ea p o . srepresents measurements of 42, 45, and 49 minutes

A good feature of these plots is that they display all the sample


values. One can reconstruct the data in its entirety from a stem-

and-leaf plot (however, the order information that items sampled is lost)


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Another Exam le: Particulate matter PM emissions for

62 vehicles driven at high altitude

Contain the a count of number of

items at or above this line

This stem contains

the medium

Contain the a count of number of

items at or below this line

Statistics-Berlin Chen 27Cumulative frequency columnStem (tens digits)

Leaf (ones digits)


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Dotplot

A dot lot is a ra h that can be used to ive a rou h

impression of the shape of a sample

Where the sample values are concentrated Where the gaps are

It is useful when the sample size is not too large and

Good method, along with the stem-and-leaf plot to

Not generally used in formal presentations

Statistics-Berlin Chen 28Figure 1.7 Dotplot of the geyser data in Table 1.3


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Histogram (1/3)

A ra h ives an idea of the sha e of a sam le

Indicate regions where samples are concentrated or sparse

The first step is to construct a frequency table

Choose boundary points for the class intervals

Compute the frequencies and relative frequencies for eachclass

Relative frequencies: frequency/sample size

Com ute the densit for each class accordin to the formula

Density = relative frequency/class width


ens y can e oug o as e re a ve requency per

unit


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Histogram (2/3)

Table 1.4A frequency table

Draw a rectangle for each class, whose height is equal to thedensity

The total areas of rectangles is equal to 1

Figure 1.8



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Histogram (3/3)

A common rule of thumb for constructin the histo ram

of a sample

It is good to have more intervals rather than fewer But it also to good to have large numbers of sample points in the

intervals

judgment and of trial and error It is reasonable to take the number of intervals roughly equal

o e square roo o e samp e s ze



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Histogram with Equal Class Widths

Default settin of most software acka e

Example: an histogram with equal class widths for Table

.

The total areas of rectangles is equal to 1

Figure 1.9

few (7) data points

Compared to Figure 1.9, Figure 1.8 presents a smoother


appearance and better enables the eye to appreciate the

structure of the data set as a whole


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Histogram, Sample Mean and Sample Variance (1/2)

ii

i valClassInterDensityOflassIntervaCenterOfCl

An approximation to the sample mean

E. ., the center of mass of the histo ram in Fi ure 1.8 is

730.6065.020177.04194.02

While the sample mean is 6.596

The narrower the rectangles (intervals), the closer the

=>items of the same value)

1, 1, 1, 2, 3, 422.1

18

22

16

14

36

52

0.5 3.5 4.5


1, 1, 1, 2, 3, 4

4.50.5 1.5 2.5 3.5

26

12

6

1

46

1

36

1

26

3

1


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Histogram, Sample Mean and Sample Variance (2/2)

Definition: The moment of inertia () for the entire

histogram is

i ramssOfHistogCenterOfMalassIntervaCenterOfCl -2

iallassIntervDensityOfC

An approximation to the sample variance

E.g., the moment of inertia for the entire histogram in Figure 1.8 is

While the sam le mean is 20.42

25.20065.0730.620177.0730.64194.0730.62 222

The narrower the rectangles (intervals) are, the closer the

approximation is



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Symmetry and Skewness (1/2)

A histo ram is erfectl s mmetric if its ri ht half is a

mirror image of its left half

E.g., heights of random men

Histograms that are not symmetric are referred to as

skewed

A histogram with a long right-hand tail is said to be

s ewe o e r g , orpos ve y s ewe

E.g., incomes are right skewed (?)

-skewed to the left, ornegatively skewed

Grades on an eas test are left skewed ?



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Symmetry and Skewness (2/2)

skewed to the left nearly symmetric skewed to the right

ere s a so ano er erm ca e ur os s a s a so

widely used for descriptive statistics

,the distribution of a population



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More on Skewness and Kurtosis (1/3)

Skewness can be used to characterize the s mmetr of

a data set (sample)

Given a sample : nXXX ,,, 21

3n Skewness is defined by

31

1 snSkewness i i

symmetric distribution shape => 0Skewnessi

: Skewed to the right0Skewness

: ewe o e e


0Skewness


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kurtosis can be used to characterize the flatness of a

data set (sample)

Given a sample : nXXX ,,, 21

Kurtosis is defined by

4

1

4XX

Kurtosisni i

A standard normal distribution has 3Kurtosis

A larger kurtosis value indicates a peaked distribution

A smaller kurtosis value indicates a flat distribution



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standard normal


http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm


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Unimodal and Bimodal Histograms (1/2)

Definition: Mode

Can refer to the most frequently occurring value in a sample

Or refer to a peak or local maximum for a histogram or othercurves

A unimodal histogram A bimodal histogram

mo a s ogram, n some cases, n ca es a e

sample can be divides into two subsamples that differ



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Unimodal and Bimodal Histograms (2/2)

Example: Durations of dormant periods (in minutes) and

lon : more than 3 minutes

short: less than 3 minutes


Histogram of all 60 durations Histogram of the durations

following short eruption

Histogram of the durationsfollowing long eruption


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Histogram with Height Equal to Frequency

Till now, we refer the term histo ram to a ra h in

which the heights of rectangles represent densities

,of rectangles equal to the frequencies

.

the heights equal to the frequencies

cf. Figure 1.8Figure 1.13

exaggerate the proportion

of vehicles in the intervals



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Boxplot (1/4)

A box lot is a ra h that resents the median, the first

()1.5 IQR

and third quartiles, and any outliers present in the

sample

1.5 IQR

1.5 IQR

The interquartile range (IQR) is the difference between the third


an rs quar e. s s e s ance nee e o span e m e

half of the data


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Boxplot (2/4)

Ste s in the Construction of a Box lot

Compute the median and the first and third quartiles of the

sample. Indicate these with horizontal lines. Draw vertical lines

Find the largest sample value that is no more than 1.5 IQR

above the third quartile, and the smallest sample value that is

not more than 1.5 IQR below the first quartile. Extend vertical

lines (whiskers) from the quartile lines to these points

1.5 IQR below the first quartile are designated as outliers. Plot

each outlier individually



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Boxplot (3/4)

Exam le: A box lot for the e ser data resented in

Table 1.5

Notice there are no outliers in these data The sample values are comparatively

densely packed between the median and

the third uartile

The lower whisker is a bit longerthan the upper one, indicating that

tail than an upper tail

The distance between the first quartile and the median is greater

than the distance between the median and the third quartile

This boxplot suggests that the data are skewed to the left (?)


B l t (4/4)


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Boxplot (4/4)

Another Exam le: Com arative box lots for PM

emissions data for vehicle driving at high versus low

altitudes


S tt l t (1/2)


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Scatterplot (1/2)

Data for which item consists of a air of values is called

bivariate

The graphical summary for bivariate data is a scatterplot Display of a scatterplot (strength of Titanium () welds

vs. its chemical contents)


S tt l t (2/2)


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Scatterplot (2/2)

Exam le: S eech feature sam le Dimensions 1 & 2 of

male (blue) and female (red) speakers after LDA

transformation

0

.

-0.4

-0.2

e

2

-0.8

-0.6featur

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2-1.2

-1

feature 1


S


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Summary

We discussed t es of data

We looked at sam lin mostl SRS

We studied summar descri tive statistics

We learned about numeric summaries We examined graphical summaries (displays of data)


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Sampling and Descriptive Statics

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