m07-numerical summaries 1 1 department of ism, university of alabama, 1995-2003 lesson objectives ...

M07-Numerical Summaries 1 1 Department of ISM, University of Alabama, 1995-2003

Lesson Objectives

Learn when each measure of a “typical value” is appropriate.Also called “central tendency” or “location.”

Learn when each measure ofa “variation” are appropriate.Also called “scatter” or “dispersion.”

See how these measures relate to statistical inference, which will coveredlater in the course.


Statistics is the science of

• collecting

• organizing

• summarizing

• interpreting

DATA

for making decisions.


Organize / SummarizeOrganize / Summarize Data Data

GraphicalGraphical NumericalNumerical


Key Features of Data Distributions

Shape

Typical Value

Spread

Outliers

This sectioncovers thesetwo.


Measures of Location

Give “middle” or “typical” valuesor “central tendency.”

Measures of VariationDescribe “spread” or “scatter”or “dispersion” in the data.


Measures of Location

1. Meanthe “center of gravity”

of the data (histogram).


formula for mean

SampleMean =

Sum of observationsdivided by

sample size

Xi

nX =

X1 + X2 + ··· +Xn

n=


The mean is ________________

to extreme values (outliers).


2. Median - midpoint of distribution

At least half of the observations

are at or less than the median,

and at least half are

at or greater than the median.


Note: For n observations,the median is located at the

n + 1

2

in the ordered sample.

-th observation


Example 1

Data: 14, 18, 20, 12, 24, 15, 14 (n = 7 “odd”)

7 + 12

= 4th location of median

Median is the middle value of the “ordered” data.

At least half the values are at or greater; at least half are at or lower.


median example

Data: 14, 18, 20, 12, 24, 15, 14 (n = 7 “odd”) 94 (outlier)

Original, Original, X = X = with outlier, X =with outlier, X =

Example 2

Median is still the middle value.

Median is resistant to outliers.


Data: 14, 18, 20, 12, 24, 15, 14, 214 (n = 8 “even,” outlier)

Median is the average of the two middle values.

Exactly half the values are greater, half lower.

Example 3

8 + 12

= 4.5th location of median


1. Order the data.

2. For odd n, the median is the center observation.

3. For even n, the median is the average of the two center observations.

Summary for finding Median


3. Mode - most frequently occurring number

In a histogram, modal class is the one having largest frequency,

i.e., highest bar.


If categorical, use the mode. “Average” is meaningless; look at “percentages” of occurrences.

If variable is quantitative, first look at a graph:

Skewed or outliers? Skewed or outliers?

More or less symmetric? More or less symmetric?

Use medianUse median..

Use meanUse mean..

When should each estimator be used?

What type of variable is it?


Numerical Summary

Location Variation

MeanMedianMode

RangeStd. DeviationIQR


Mountain Climbing Rope.Two suppliers; sample and

test three ropes from each.

“Snap Breaking Strength”

Why does variation matter?


Measures of Variation

1. Range

2. Variance & Standard Deviation

3. Mean Absolute Deviation (Mad)

4. Interquartile Range (IQR)


Highest minus lowest value in the sample.

1. Range


Example 4: 3, 4, 1, 7, 4,

5

1 2 3 4 5 6 7

Example 5: 1, 1, 1, 7, 7,

7

1 2 3 4 5 6 7

Range =

Range =


Advantage: _________

_________________.

Disadvantage:

_______ most of the data.

______________ to outliers.

Range


How far are the data from the middle, on average?

2. Variance & Standard Deviation

Sample Variance = s2

Sample Std. Dev. = sPopulation Variance = 2

Population Std. Dev. =

Notation:


Example 4: 3, 4, 1, 7, 4, 5

1 2 3 4 5 6 7


We need to keep the negatives from canceling the positives.

We can do this by 1. _____________, ______

2. _____________, _____

Note: The average of the deviations from the mean will always be zero.


Equation for Variance:

2 = N

(Xi - )2

(see page 88)

For a population:

s2 = n - 1

(Xi - X)2

For a sample:


Equation for Variance:

s2 = n - 1

(Xi - X)2(see page 88)

=

= units?

Example 4 data:

(3-4)2 + (4-4)2 + (1-4)2 + (7-4)2 + (4-4)2 + (5-4)2

6 - 1=


Equations for Variance:

(see page 88)s2 = n - 1

(Xi - X)2

(see page 90)sX Xn

n 12 i

2 2 =

sX

( X )n

n 12 i

2 i2

=

1.

2.

3.


Example 4: 3, 4, 1, 7, 4, 5X3417

4 524

X9

161

491625

116

2

X =

X2 =

X =

X2 =


sX

( X )n

n 12

i2 i

2

=s2

6 - 1= 4.0


• Both equations shouldgive the same answer.

• First is easier when data and the mean are integers.

• Second is easier for larger data sets, or data not integer.

• More chance of round-off errorwith first equation.

Comments


Advantage: ________________; ________________.

Disadvantages: Units are _________.

____ resistant to outliers.

Variance


Standard DeviationS = S

2 “The square root of the variance.”

= 4.0

= 2.0

Advantage: Easier to interpret than variance, Units same as data.


3. Mean Absolute Deviation, MAD

MAD = xi – N

This will be used extensively in OM 300

for population data

MAD = xi – x n

for sample data

(see page 87)


IQR = Q - Q 13

IQR is the range of the middle 50% of the data.

Observations more than 1.5 IQR’s beyond quartiles are considered outliers.

4. Interquartile Range (IQR)

Mor

e on

this

late

r.

C07-Numerical Summaries 1 36 Department of ISM, University of Alabama, 1995-2003

Statistical Inference

Generalizing from a sample to a population,

by using a statisticto estimate

a parameter.


ParameterParameterStatisticStatistic

Mean:

Standard deviation:

Proportion:

s

X

estimates

estimates

estimatesp

from sample

from entire population


Descriptive

NumericalGraphical

Statistics


Estimate the true mean net weight of Estimate the true mean net weight of 16 oz. bags of Golden Flake Potato Chips 16 oz. bags of Golden Flake Potato Chips with a 95% confidence interval. with a 95% confidence interval.

16.0516.0115.9215.6816.1016.0115.7215.8016.2115.70

15.9516.2416.0215.9016.0716.0516.1815.4516.0416.05

Measured Weights in ounces.

Use MinitabUse Minitab

Is the filling machinedoing what it shouldbe doing?

Is the filling machinedoing what it shouldbe doing?

(Not real (Not real

data)data)

Example 5:


Data window

name of worksheet file

Most commonlyused features.

Session window


“Stat”

“Basic Statistics ”

“Display descriptive statistics”


Results for: c07 Weight of chips.MTW

Descriptive Statistics: Weights

Variable N Mean Median TrMean StDev SE Mean

Weights 20 15.958 16.015 15.970 0.199 0.045

Variable Minimum Maximum Q1 Q3

Weights 15.450 16.240 15.825 16.065

Executing from file: C:\Program Files\MTBWIN\MACROS\Describe.MAC

Descriptive Statistics Graph: Weights

“Session Window” results“Session Window” results

“Five number” summary

Histogram with Normal distribution

curve superimposed

Box plot

“95% Confidence Interval”

for the population mean.

____, because 16.000 is a plausible value for the truepopulation mean.

____, because 16.000 is a plausible value for the truepopulation mean.

A confidence interval gives the limits of the plausible values of the true population mean, .

A confidence interval gives the limits of the plausible values of the true population mean, .

Our sample mean was 15.957 oz.This is less than 16.000.Should we be concerned?

Our sample mean was 15.957 oz.This is less than 16.000.Should we be concerned?

“95% Confidence Interval”

for the population mean.

m07-numerical summaries 1 1 department of ism, university of alabama, 1995-2003 lesson objectives ...

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