statistics sampling and methods wbhs

8/6/2019 Statistics Sampling and Methods WBHS

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A focus on Sampling andA focus on Sampling and

Sampling MethodsSampling Methods


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MenuMenuSampling Methods

Measures of Centre

Measures of Spread

Definitions

Assessment Tips

Practice Tasks

For clarification, click on any step you do not understand to see that

element broken down

The example used throughout this presentation is trying to find the

mean height of WBHS pupils

On Your Calculator


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Sampling MethodsSampling Methods

In this presentation youIn this presentation you

will see a number ofwill see a number of

sampling methods, theirsampling methods, theirbenefits and drawbacks.benefits and drawbacks.

Simple Random Sample

Cluster Sampling

Systematic Sampling

Stratified Sampling

Note: For more detailed instructions

on any of the example click on

the step you misunderstand


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Measures of Central TendencyMeasures of Central Tendency

In this presentationIn this presentation

you will learn how toyou will learn how to

calculate a number ofcalculate a number ofmeasures of averagemeasures of average

or centre, as well asor centre, as well as

their benefits andtheir benefits and

drawbacksdrawbacks

Mean

Median

Mode


in any of the examples click

on the step you misunderstand


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Measures ofSpreadMeasures ofSpread

In this presentation youIn this presentation you

will learn how to find awill learn how to find a

number of measures ofnumber of measures of

spread as well as theirspread as well as their

drawbacks and advantages.drawbacks and advantages.

You will also need toYou will also need to

decide which measure ofdecide which measure ofspread and which measurespread and which measure

of centre go together.of centre go together.

Standard Deviation

Interquartile Range

Range


in any of the examples click

on the step you misunderstand


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Simple RandomSampleSimple RandomSample

The simplest unbiasedThe simplest unbiased

sample.sample.

11-- Number the entireNumber the entire

population.population.

22-- Generate randomGenerate random

numbers.numbers.

33-- Proceed until you haveProceed until you haveas many as you needas many as you need

ignoring any repeats.ignoring any repeats.

Example (Heights of WBHS students)Example (Heights of WBHS students)

1.1. Get a copy of the School Roll.Get a copy of the School Roll.

2.2. Number every personNumber every person

3.3. Generate Random numbers from 1Generate Random numbers from 1

to the maximum you need.to the maximum you need.

4.4. Proceed until you have the desiredProceed until you have the desired

sample size ignoring repeats.sample size ignoring repeats.


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AdvantagesAdvantages

CheapCheap

Easy to carry outEasy to carry outUnbiasedUnbiased

DisadvantagesDisadvantages

May not represent strataMay not represent strata

Needs an entire populationNeeds an entire populationlistlist


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Cluster SamplingCluster Sampling

The easiest unbiasedThe easiest unbiased

sample.sample.

1.1. Sort your data intoSort your data into

clusters based on location.clusters based on location.

2.2. Randomly choose theRandomly choose the

cluster.cluster.

3.3. Perform a simple randomPerform a simple randomsample on the chosensample on the chosen

cluster.cluster.



2.2. Sort into clustersSort into clusters egeg year levelsyear levels

3.3. Randomly select the cluster.Randomly select the cluster.

4.4. Randomly generate a sample fromRandomly generate a sample from

each cluster.each cluster.

Care with clusters as Juniors areCare with clusters as Juniors aremuch shorter than Seniorsmuch shorter than Seniors


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Cluster SamplingCluster Sampling


Very CheapVery Cheap

Very Easy to carry outVery Easy to carry outUnbiasedUnbiased


Needs an entire populationNeeds an entire population

listlist

Can be biased if clustersCan be biased if clusters

strongly affect thestrongly affect the

statistics.statistics.


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Systematic SamplingSystematic Sampling

A relatively quick way toA relatively quick way to

pick an unbiased samplepick an unbiased sample

1.1. List the entire population.List the entire population.

2.2. Decide on your step sizeDecide on your step size

(Total(Total Sample size = n)Sample size = n)..

3.3. Randomly generate aRandomly generate a

starting point.starting point.4.4. Step every nStep every nthth data pointdata point

till you have your sample.till you have your sample.


1.1. Get an alphabetical copy of theGet an alphabetical copy of the

School Roll.School Roll.

2.2. StepSize = TotalStepSize = Total Sample sizeSample size

3.3. Randomly generate a starting point.Randomly generate a starting point.

4.4. Starting from the beginning use theStarting from the beginning use the

step size to pick the rest of thestep size to pick the rest of the

samplesample


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Systematic SamplingSystematic Sampling


CheapCheap

Easy to Choose SampleEasy to Choose SampleUnbiasedUnbiased


Needs an entire populationNeeds an entire population

listlist

If population list is orderedIf population list is ordered

then sample can becomethen sample can become

biasedbiased


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Stratified SamplingStratified Sampling

The most reliableThe most reliablesampling method.sampling method.

1.1. Sort the data into strataSort the data into strata

based on information youbased on information youalready know.already know.

2.2. Calculate the proportionsCalculate the proportionsfor each strata.for each strata.

3.3. Perform a Simple RandomPerform a Simple RandomSample on each of theSample on each of thestrata.strata.


1.1. Get a copy of the School RollGet a copy of the School Roll

separated into year levels.separated into year levels.

2.2. Calculate the sample size for eachCalculate the sample size for each

year group (strata).year group (strata).

3.3. Performa simple random samplePerforma simple random sample

on each year group to their specificon each year group to their specific

sample size.sample size.


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Stratified SamplingStratified Sampling


UnbiasedUnbiased

CompletelyCompletelyrepresentative of eachrepresentative of each

of the strataof the strata

Most reliable estimatesMost reliable estimates


Needs entire populationNeeds entire population

listlist

Information about entireInformation about entire

population needs to bepopulation needs to be

known beforehandknown beforehand

Time consumingTime consuming


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Generate a Random NumberGenerate a Random Number

1.1. Decide on the startingDecide on the starting

number (in this case 1)number (in this case 1)

2.2. Decide how many youDecide how many you

need (In the case of theneed (In the case of the

school 529 students)school 529 students)

3.3. Choose your calculatorChoose your calculator

Casio

FX-82

Casio

Graphic

Texas


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Random Number on a CasioRandom Number on a Casio

Graphics CalculatorGraphics Calculator1.1. Decide on the startingDecide on the starting





3.3. InIn RunRun ModeModeIntgIntg OPTNOPTN F6F6 F4F4 F5F5

Ran#Ran# OPTNOPTN F6F6 F3F3 F4F4

On ScreenOn Screen

IntgIntg(529(529 Ran# + 1)Ran# + 1)

Population sizeor Strata size Starting Value

OPTN

F3 F4 F6

( )

7 8

5

+1

Intg(529 Ran# + 1)


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Random Number on a Casio FXRandom Number on a Casio FX -- 8282






3.3. Ran# = 2Ran# = 2ndnd functionfunction

4.4. On screenOn screen

Ran#Ran# 529 + 1 =529 + 1 =

notenote Ignore any decimal in theIgnore any decimal in the

answeranswer

Population size

or strata size

Starting value

RAN#529+1

shift


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Random Number on a TexasRandom Number on a Texas






RANDIRANDI PRBPRB RANDIRANDI

,, 22ndnd Function )Function )

3.3. On ScreenOn Screen

RANDI(1 , 529)RANDI(1 , 529)

End ValueStarting Value

2nd

PRB

)

RANDI(1,529)


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The simplest unbiasedThe simplest unbiasedsample.sample.

1.1. Number the entireNumber the entire

population.population.2.2. Generate randomGenerate random

numbers.numbers.

3.3. Proceed until youProceed until you

have as many as youhave as many as youneed ignoring anyneed ignoring anyrepeats.repeats.



2.2. Number every person from 1 (toNumber every person from 1 (to

529)529)

3.3. Generate Random numbers from 1Generate Random numbers from 1

to the maximum you need (529).to the maximum you need (529).

4.4. Proceed until you have the desiredProceed until you have the desired

sample size ignoring repeats.sample size ignoring repeats.


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Strata ProportionsStrata Proportions

1.1. Number of peopleNumber of people

in strata divided byin strata divided by

total in population.total in population.

2.2. Multiplied byMultiplied by

number of peoplenumber of people

wanted in totalwanted in total

sample.sample.


1.1. 529 people on School Roll.529 people on School Roll.

2.2. 115 year 10s115 year 10s

3.3. Sample size of 30Sample size of 30

4.4. So year 10 sample sizeSo year 10 sample size

115115 529529 30 = 6.5230 = 6.52

So take 7 year 10 studentsSo take 7 year 10 students


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Systematic StepSizesSystematic StepSizes

1.1. Number of peopleNumber of people

in populationin population

divided by Sampledivided by Sample

SizeSize


1.1. 529 people on School Roll.529 people on School Roll.

2.2. Sample size of 30Sample size of 30

3.3. So Step sizeSo Step size

529529 30 = 17.6333330 = 17.63333

So take every 17So take every 17thth student from thestudent from the

starting positionstarting position


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Systematic SteppingSystematic Stepping

1.1. Starting at theStarting at the

random start pointrandom start point

step out till you getstep out till you get

desired sample size.desired sample size.


1.1. Random starting point 803, stepRandom starting point 803, step

size 29size 29

2.2. 803803rdrd student on alphabeticallist isstudent on alphabeticallist is

where we start.where we start.

3.3. Then 832Then 832ndnd student, 861student, 861stst student,student,

we have now reached the end of thewe have now reached the end of the

roll so start at the beginning 890=roll so start at the beginning 890=1515thth student then 45student then 45thth studentstudent


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MeanMean1.1. Add up all of theAdd up all of the

values in the sample.values in the sample.

2.2. Divide by the sampleDivide by the sample

size.size.

Advantages

Easy to calculate for large

samples.

Accurate and well understood

Disadvantages

Affected by outliers

Calculator Method


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MedianMedian

1.1. List all the values inList all the values in

order.order.

2.2. Find the central valueFind the central value

Advantages

Accurate

Not affected much by Outliers

Disadvantages

Not so widely known as an

average

Time consuming to list large

sample in order


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ModeMode

1.1. List all the valuesList all the values

2.2. Find the most commonFind the most common

itemitem

Advantages

Can calculate mode for data that

is not numeric or ordered

Not affected much by Outliers

Very easy to calculate

Disadvantages

Can be inaccurate for numeric or

data that can be ordered


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Statistics on a CalculatorStatistics on a Calculator

Choose your calculatorChoose your calculator

Casio

FX-82

Casio

Graphic

Texas


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1.1. In Stat ModeIn Stat Mode

2.2. InIn list 1list 1 enter all data valuesenter all data values

3.3. InIn list 2list 2 enter theirenter their

frequenciesfrequencies

4.4. F2 (CALC)F2 (CALC)

5.5. F6 (SET)F6 (SET) Should readShould read

6.6. ExitExit

7.7. F1 (1VAR)F1 (1VAR)(All Statistics are listed(All Statistics are listed is meanis mean,, n is std. devn is std. dev.).)

F2F1 F6

1Var XList :List1

1Var Freq :List2

2Var XList :List3

2Var YList :List4

2Var Freq :List5

EXIT

Statistics on aStatistics on a

Casio Graphics CalculatorCasio Graphics Calculator

S.D. using table


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Entering Data onEntering Data on

Casio Graphics CalculatorCasio Graphics CalculatorEnter each data valuein List 1 followed by

EXE

Enter the frequency of

each data value in List 2

followed by EXE

Note If all of the frequencies are

1 then you dont need to

enter the frequencies.

In the Set Menu change the 1Var Freq

to 1 instead of list 2

List 1 List 2 List 3 List4

1

2

3

4

5

EXE


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1.1. Put your calculator intoPut your calculator into

statistics modestatistics mode Mode 2Mode 2

2.2. Clear the statistics memoryClear the statistics memory Shift Mode 1Shift Mode 1

3.3. Enter the data carefullyEnter the data carefully 180cm M+180cm M+

4.4. Calculate desired statisticsCalculate desired statistics Shift 2Shift 2

1.1. meanmean

2.2. nn standard deviationstandard deviation


Casio FX 82 CalculatorCasio FX 82 CalculatorScl mode clr all

1 2 3

shift mode

M+

Shown on Screen

S.D. using table


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Casio FX 82 CalculatorCasio FX 82 Calculator

Enter each data value

followed by M+

n is the number of data

values that you have entered

Note Be very careful entering thedata values as you cannot

review them later to make

sure that they are correct.

n =

1

M+


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Texas CalculatorTexas Calculator1.1. Put your calculator intoPut your calculator intostatistics modestatistics mode

1.1. 22ndnd Function DATAFunction DATA

2.2. 11 -- VARVAR

2.2. Enter the data carefullyEnter the data carefully

1.1. DATADATA

3.3. Calculate desired statisticsCalculate desired statistics

1.1. STATVARSTATVAR

2.2. Shift between statistics with arrowShift between statistics with arrowkeyskeys

1.1. nn number of data valuesnumber of data values

2.2. meanmean

3.3. standard deviationstandard deviation

S.D. using table

2nd

DATA

n x Sx x

STATVAR


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Entering Data on aEntering Data on a

Texas CalculatorTexas Calculator

2nd

DATA

X1 = 180

Press the Data Key

to begin

Begin entering data.

X1 is the data value

Followed by the down arrow

Freq1 is that data values frequencyFollowed by the down arrow

X2 is next then Freq2

To check data use up arrow


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DefinitionsDefinitions

PopulationPopulation The entire list of those people or things that you wishThe entire list of those people or things that you wishto sampleto sample

CensusCensus A survey of anA survey of an entire populationentire population

SampleSample A small group of a populationA small group of a population

ParametersParameters Facts about an entire populationFacts about an entire population gained from a censusgained from a census

(Notation: mean (Notation: mean or standard deviation or standard deviation ))

StatisticsStatistics Estimates of population parametersEstimates of population parameters calculated from acalculated from asamplesample

(Notation: mean (Notation: mean or standard deviation s) or standard deviation s)

RepresentativeRepresentative A sample that appears toA sample that appears to represent all elements of therepresent all elements of the

in the correct proportionsin the correct proportionspopulationpopulation BiasBias A sampling method thatA sampling method that does not give every element ofdoes not give every element of

the population an equal chance of selectionthe population an equal chance of selection


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Standard DeviationStandard Deviation This is a calculation of theThis is a calculation of the

average difference betweenaverage difference betweenthe data values and thethe data values and themean.mean.

This measure of spreadThis measure of spreadapplies to the mean.applies to the mean.

Advantages


samples on calculator.

Accurate

Very useful for certain types of

data

Disadvantages

Affected by outliers

Possibly not so well understood

Use Calculatorto Calculate Use table to calculate


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Interquartile RangeInterquartile Range1.1. Calculate the upper andCalculate the upper and

lower quartiles.lower quartiles.

2.2. Upper quartile minus lowerUpper quartile minus lowerquartile.quartile.

3.3. This measure of spreadThis measure of spreadapplies to the medianapplies to the median

Advantages

Well understood

Unaffected by outliers

Disadvantages


samples.


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1.1. Find the highest and lowestFind the highest and lowestvalue.value.

2.2. Highest value minus theHighest value minus thelowest value.lowest value.

3.3. This measure of spreadThis measure of spreadapplies to all measures ofapplies to all measures ofcentre.centre.

RangeRange

Advantages

Well understood

Unaffected by outliers

Disadvantages


samples.


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Standard Deviation by TableStandard Deviation by Table

( )2

180 165 15 225

150 165 -15 225

165 165 0 0

170 165 5 25

160 165 -5 25

Total 825 0 500

Mean 165 100

Data Values

From your sample

or census

Mean

Calculated as usual,

doesnt change Data values minus the Mean

Square of each ofthe values to the left

Final Standard Deviation

is the square root of this

value so s = 10

Use Calculator to Calculate


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1.1. List all the values in order.List all the values in order.

2.2. Find the central valueFind the central value

3.3. Discard that central valueDiscard that central value

4.4. Find the central value of theFind the central value of the

remaining two halves.remaining two halves.

5.5. These 2 numbers are theThese 2 numbers are the

upper and lower quartilesupper and lower quartiles

Calculating QuartilesCalculating Quartiles


1.1. Data ValuesData Values

165, 170, 173, 180, 182, 183, 191, 192165, 170, 173, 180, 182, 183, 191, 192

2.2. Central value middle of 180 and 182Central value middle of 180 and 182

so median is 181so median is 181

3.3. Discard 181 and calculate middle ofDiscard 181 and calculate middle of

each half.each half.

4.4. 165, 170, 173, 180//182, 183, 191, 192165, 170, 173, 180//182, 183, 191, 192

Lower quartile Upper quartileLower quartile Upper quartile

171171 187187


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Things to ConsiderThings to Consider

Is my sample representative of the population?

Need to consider whether any strata present in the data are

represented in approximately the correct proportions.

Need to consider the presence of any apparent outliers in the sample

chosen, and the effect they will have on estimates of population

parameters.


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Is my sample representative of the population?

Estimates are more reliable when taken from a large sample as the

effects of outliers are lessened.

Consider the size of the s.d.

A larger value of s suggests considerable variation in the data

values. Thus taking another sample could produce quite different

statistics.

Ask yourself, IfI were to repeat this sampling process, would I getthe same results?


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How could I improve my sampling method?

Need to choose a sampling method which eliminates bias, and which

gives the best chance of choosing a representative sample. (Bias

exists when some of the population members have greater or lesser

chance of being included in the sample.)

Need to discuss which statistics would give the best estimates of

population parameters, including the effect of outliers.


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Would I get the same or similar results if I repeated

the same process?

Are there outliers or extreme values that may affect the samplestatistics? If so then I probably wouldnt get similar results.

Is the standard deviation (or measure of spread) large when

compared to the mean, if it is then repeating the same results is

unlikely.


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When answering question or stating conclusions;

Answers need to be precise and refer to actual data values present in

the sample and/or population.

Strata must be clearly defined.

Answers cannot be vague or rote-learnt without referring specifically

to the context of the assessment.

Students must be very clear that the sample statistics are ESTIMATES

of the population parameters.

They must NOT state that the population mean is unless they have

taken a census of the whole population!


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Practice TasksPractice Tasks

Real Estate Stats


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On Your CalculatorOn Your Calculator

In this part of theIn this part of the

presentation you canpresentation you can

check on exactly howcheck on exactly how

to use your calculatorto use your calculator

effectively to help witheffectively to help with

StatisticsStatistics

Generating Random Numbers

Entering Data

Calculating Statistics


on any of the example click on

the step you misunderstand


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Entering Data on a CalculatorEntering Data on a Calculator


Casio

FX-82

Casio

Graphic

Texas


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Statistics on a CalculatorStatistics on a Calculator


Casio

FX-82

Casio

Graphic

Texas


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1.1. In Stat ModeIn Stat Mode

2.2. InIn list 1list 1 enter all data valuesenter all data values

3.3. InIn list 2list 2 enter theirenter their

frequenciesfrequencies

4.4. F2 (CALC)F2 (CALC)

5.5. F6 (SET)F6 (SET) Should readShould read

6.6. ExitExit7.7. F1 (1VAR)F1 (1VAR)(All Statistics are listed(All Statistics are listed is meanis mean,, n is std. devn is std. dev.).)

F2F1 F6

1Var XList :List1

1Var Freq :List2

2Var XList :List3

2Var YList :List4

2Var Freq :List5

EXIT


Casio Graphics CalculatorCasio Graphics Calculator

S.D. using table


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Casio Graphics CalculatorCasio Graphics CalculatorEnter each data valuein List 1 followed by

EXE

Enter the frequency of

each data value in List 2

followed by EXE

Note If all of the frequencies are

1 then you dont need to

enter the frequencies.

In the Set Menu change the 1Var Freq

to 1 instead of list 2

List 1 List 2 List 3 List4

1

2

3

4

5

EXE


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1.1. Put your calculator intoPut your calculator into

statistics modestatistics mode Mode 2Mode 2

2.2. Clear the statistics memoryClear the statistics memory Shift Mode 1Shift Mode 1

3.3. Enter the data carefullyEnter the data carefully 180cm M+180cm M+

4.4. Calculate desired statisticsCalculate desired statistics Shift 2Shift 2

1.1. meanmean

2.2. nn standard deviationstandard deviation


Casio FX 82 CalculatorCasio FX 82 CalculatorScl mode clr all

1 2 3

shift mode

M+

Shown on Screen

S.D. using table


50/56


Casio FX 82 CalculatorCasio FX 82 Calculator

Enter each data value

followed by M+

n is the number of data

values that you have entered

Note Be very careful entering thedata values as you cannot

review them later to make

sure that they are correct.

n =

1

M+


51/56


Texas CalculatorTexas Calculator1.1. Put your calculator intoPut your calculator into

statistics modestatistics mode

1.1. 22ndnd Function DATAFunction DATA

2.2. 11 -- VARVAR

2.2. Enter the data carefullyEnter the data carefully

1.1. DATADATA

3.3. Calculate desired statisticsCalculate desired statistics

1.1. STATVARSTATVAR

2.2. Shift between statistics with arrowShift between statistics with arrowkeyskeys

1.1. nn number of data valuesnumber of data values

2.2. meanmean

3.3. standard deviationstandard deviation

S.D. using table

2nd

DATA

n x Sx x

STATVAR


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Entering Data on aEntering Data on a

Texas CalculatorTexas Calculator

2nd

DATA

X1 = 180

Press the Data Key

to begin

Begin entering data.

X1 is the data value

Followed by the down arrow

Freq1 is that data values frequencyFollowed by the down arrow

X2 is next then Freq2

To check data use up arrow


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Generate a Random NumberGenerate a Random Number




need (In the case of theneed (In the case of theschool 529 students)school 529 students)

3.3. Choose your calculatorChoose your calculator

Casio

FX-82

Casio

Graphic

Texas


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Random Number on a CasioRandom Number on a Casio

Graphics CalculatorGraphics Calculator





3.3. InIn RunRun ModeModeIntgIntg OPTNOPTN F6F6 F4F4 F5F5

Ran#Ran# OPTNOPTN F6F6 F3F3 F4F4

On ScreenOn Screen

IntgIntg(529(529 Ran# + 1)Ran# + 1)

Population size

or Strata size Starting Value

OPTN

F3 F4 F6

( )

7 8

5

+1

Intg(529 Ran# + 1)


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Random Number on a Casio FXRandom Number on a Casio FX -- 8282






3.3. Ran# = 2Ran# = 2ndnd functionfunction

4.4. On screenOn screen

Ran#Ran# 529 + 1 =529 + 1 =

notenote Ignore any decimal in theIgnore any decimal in the

answeranswer

Population size

or strata size

Starting value

RAN#529+1

shift


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Random Number on a TexasRandom Number on a Texas





RANDIRANDI PRBPRB RANDIRANDI

,, 22ndnd Function )Function )3.3. On ScreenOn Screen

RANDI(1 , 529)RANDI(1 , 529)

E d V lSt ti V l

2nd

PRB

)

RANDI(1,529)

statistics sampling and methods wbhs

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