A focus on Sampling and Sampling Methods

A focus on Sampling andA focus on Sampling and

Sampling MethodsSampling Methods

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

Sampling MethodsSampling Methods

In this presentation you In this presentation you will see a number of will see a number of sampling methods, their sampling methods, their benefits 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

Measures of Central TendencyMeasures of Central Tendency

In this presentation In this presentation you will learn how to you will learn how to calculate a number of calculate a number of measures of average measures of average or centre, as well as or centre, as well as their benefits and their benefits and drawbacksdrawbacks

Mean

Median

Mode

Note: For more detailed instructions in any of the examples click

on the step you misunderstand

Measures of SpreadMeasures of Spread

In this presentation you In this presentation you will learn how to find a will learn how to find a number of measures of number of measures of spread as well as their spread as well as their drawbacks and advantages.drawbacks and advantages.

You will also need to You will also need to decide which measure of decide which measure of spread and which measure spread and which measure of centre go together.of centre go together.

Standard Deviation

Interquartile Range

Range

Note: For more detailed instructions in any of the examples click

on the step you misunderstand

Simple Random SampleSimple Random Sample

The simplest unbiased The simplest unbiased sample. sample.

1-1- Number the entire Number the entire population.population.

2-2- Generate random Generate random numbers.numbers.

3-3- Proceed until you have Proceed until you have as many as you need as 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 1 Generate Random numbers from 1 to the maximum you need.to the maximum you need.

4.4. Proceed until you have the desired Proceed until you have the desired sample size ignoring repeats.sample size ignoring repeats.


AdvantagesAdvantages

Cheap Cheap

Easy to carry outEasy to carry out

UnbiasedUnbiased

DisadvantagesDisadvantages

May not represent strataMay not represent strata

Needs an entire population Needs an entire population listlist

Cluster SamplingCluster Sampling

The easiest unbiased The easiest unbiased sample. sample.

1.1. Sort your data into Sort your data into clusters based on location.clusters based on location.

2.2. Randomly choose the Randomly choose the cluster.cluster.

3.3. Perform a simple random Perform a simple random sample on the chosen sample on the chosen cluster.cluster.



2.2. Sort into clusters eg year levelsSort into clusters eg year levels

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

4.4. Randomly generate a sample from Randomly generate a sample from each cluster.each cluster.

Care with clusters as Juniors are Care with clusters as Juniors are much shorter than Seniorsmuch shorter than Seniors

Cluster SamplingCluster Sampling


Very Cheap Very Cheap

Very Easy to carry outVery Easy to carry out

UnbiasedUnbiased



Can be biased if clusters Can be biased if clusters strongly affect the strongly affect the

statistics. statistics.

Systematic SamplingSystematic Sampling

A relatively quick way to A relatively quick way to pick an unbiased sample pick an unbiased sample

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

2.2. Decide on your step size Decide on your step size (Total (Total ÷ Sample size = n)÷ Sample size = n)..

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

4.4. Step every nStep every nthth data point data point till you have your sample.till you have your sample.


1.1. Get an alphabetical copy of the Get an alphabetical copy of the School Roll.School Roll.

2.2. Step Size = Total Step Size = Total ÷ Sample size÷ Sample size

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

4.4. Starting from the beginning use the Starting from the beginning use the step size to pick the rest of the step size to pick the rest of the samplesample

Systematic SamplingSystematic Sampling


Cheap Cheap

Easy to Choose SampleEasy to Choose Sample

UnbiasedUnbiased



If population list is ordered If population list is ordered then sample can become then sample can become

biasedbiased

Stratified SamplingStratified Sampling

The most reliable The most reliable sampling method. sampling method.

1.1. Sort the data into strata Sort the data into strata based on information you based on information you already know.already know.

2.2. Calculate the proportions Calculate the proportions for each strata.for each strata.

3.3. Perform a Simple Random Perform a Simple Random Sample on each of the Sample on each of the strata.strata.


1.1. Get a copy of the School Roll Get a copy of the School Roll separated into year levels.separated into year levels.

2.2. Calculate the sample size for each Calculate the sample size for each year group (strata).year group (strata).

3.3. Perform a simple random sample Perform a simple random sample on each year group to their specific on each year group to their specific sample size. sample size.

Stratified SamplingStratified Sampling

Advantages Advantages

UnbiasedUnbiased

Completely Completely representative of each representative of each

of the strataof the strata

Most reliable estimatesMost reliable estimates


Needs entire population Needs entire population listlist

Information about entire Information about entire population needs to be population needs to be

known beforehandknown beforehand

Time consumingTime consuming

Generate a Random NumberGenerate a Random Number

1.1. Decide on the starting Decide on the starting number (in this case 1)number (in this case 1)

2.2. Decide how many you Decide how many you need (In the case of the need (In the case of the school 529 students)school 529 students)

3.3. Choose your calculatorChoose your calculator

Casio

FX-82

Casio

GraphicTexas

Random Number on a Casio Random Number on a Casio Graphics CalculatorGraphics Calculator



3.3. In In Run Run ModeModeIntgIntg OPTN – F6 – F4 – F5 OPTN – F6 – F4 – F5

Ran#Ran# OPTN – F6 – F3 – F4 OPTN – F6 – F3 – F4

On ScreenOn Screen

Intg(529 Intg(529 × Ran# + 1)× Ran# + 1)

Population size or Strata size Starting Value

OPTN

F3 F4 F6

( )

7 8

5 ×

+1

Intg(529 × Ran# + 1)

Random Number on a Casio FX - 82Random Number on a Casio FX - 82



3.3. Ran# = 2Ran# = 2ndnd function function ··

4.4. On screenOn screen

Ran# × 529 + 1 =Ran# × 529 + 1 =

notenote Ignore any decimal in the Ignore any decimal in the answeranswer

Population size or strata size

Starting value

RAN#×529+1

·

shift

Random Number on a TexasRandom Number on a Texas



RANDIRANDI PRB PRB →→ RANDI RANDI

,, 22ndnd Function ) Function )

3.3. On Screen On Screen RANDI(1 , 529)RANDI(1 , 529)

End ValueStarting Value

2nd

PRB

)

RANDI(1,529)


The simplest unbiased The simplest unbiased sample. sample.

1.1. Number the entire Number the entire population.population.

2.2. Generate random Generate random numbers.numbers.

3.3. Proceed until you Proceed until you have as many as you have as many as you need ignoring any need ignoring any repeats.repeats.



2.2. Number every person from 1 (to Number every person from 1 (to 529)529)

3.3. Generate Random numbers from 1 Generate Random numbers from 1 to the maximum you need (529).to the maximum you need (529).

4.4. Proceed until you have the desired Proceed until you have the desired sample size ignoring repeats.sample size ignoring repeats.

Strata ProportionsStrata Proportions

1.1. Number of people Number of people in strata divided by in strata divided by total in population. total in population.

2.2. Multiplied by Multiplied by number of people number of people wanted in total wanted in total sample.sample.


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

2.2. 115 year 10’s115 year 10’s

3.3. Sample size of 30Sample size of 30

4.4. So year 10 sample sizeSo year 10 sample size

115 115 ÷ 529 × 30 = 6.52÷ 529 × 30 = 6.52

So take 7 year 10 studentsSo take 7 year 10 students

Systematic Step SizesSystematic Step Sizes

1.1. Number of people Number of people in population in population divided by Sample divided by Sample SizeSize


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

2.2. Sample size of 30 Sample size of 30

3.3. So Step sizeSo Step size

529 529 ÷ 30 = 17.63333÷ 30 = 17.63333

So take every 17So take every 17thth student from the student from the starting positionstarting position

Systematic SteppingSystematic Stepping

1.1. Starting at the Starting at the random start point random start point step out till you get step out till you get desired sample size.desired sample size.


1.1. Random starting point 803, step Random starting point 803, step size 29size 29

2.2. 803803rdrd student on alphabetical list is student on alphabetical list is where we start.where we start.

3.3. Then 832Then 832ndnd student, 861 student, 861stst student, student, we have now reached the end of the we have now reached the end of the roll so start at the beginning 890= roll so start at the beginning 890= 1515thth student then 45 student then 45thth student… student…

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

values in the sample.values in the sample.

2.2. Divide by the sample Divide by the sample size.size.

Advantages

Easy to calculate for large samples.

Accurate and well understood

Disadvantages

Affected by outliers

Calculator Method

MedianMedian

1.1. List all the values in List 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

ModeMode

1.1. List all the values List all the values

2.2. Find the most common Find 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

Statistics on a CalculatorStatistics on a Calculator

Choose your calculatorChoose your calculator

Casio

FX-82

Casio

Graphic

Texas

1.1. In Stat ModeIn Stat Mode

2.2. In In list 1list 1 enter all data values enter all data values

3.3. In In list 2list 2 enter their enter 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 χχ is mean is mean, , χσχσn is std. devn is std. dev.).)

F2F1 F6

1Var XList :List11Var Freq :List22Var XList :List32Var YList :List42Var Freq :List5

EXIT

Statistics on a Statistics on a Casio Graphics CalculatorCasio Graphics Calculator

S.D. using table

Entering Data on Entering Data on Casio Graphics CalculatorCasio Graphics Calculator

Enter each data value in 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 don’t 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 List412345

EXE

1.1. Put your calculator into Put your calculator into statistics mode statistics mode

• Mode 2Mode 2

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

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

4.4. Calculate desired statisticsCalculate desired statistics• Shift 2Shift 2

1.1. χχ mean mean

2.2. χσχσnn standard standard deviationdeviation

Statistics on a Statistics on a Casio FX 82 CalculatorCasio FX 82 Calculator

Scl mode clr all

1 2 3

shift mode

M+

Shown on Screen

S.D. using table

Entering Data on Entering Data on 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 the data values as you cannot review them later to make sure that they are correct.

n =

1

M+

Statistics on a Statistics on a Texas CalculatorTexas Calculator

1.1. Put your calculator into Put your calculator into statistics modestatistics mode

1.1. 22ndnd Function DATA Function DATA

2.2. 1 - VAR1 - VAR

2.2. Enter the data carefullyEnter the data carefully 1.1. DATADATA

3.3. Calculate desired statisticsCalculate desired statistics1.1. STATVARSTATVAR

2.2. Shift between statistics with arrow Shift between statistics with arrow keyskeys1.1. nn number of number of

data valuesdata values

2.2. χχ mean mean

3.3. σχσχ standard standard deviationdeviationS.D. using table

2nd

DATA

n x Sx σx

STATVAR

Entering Data on a Entering Data on a Texas CalculatorTexas Calculator

2nd

DATA

X1 = 180

Press the Data Key to begin

Begin entering data.X1 is the data valueFollowed by the down arrow

Freq1 is that data values frequencyFollowed by the down arrowX2 is next then Freq2To check data use up arrow

DefinitionsDefinitions• PopulationPopulation The entire list of those people or things that you wish The entire list of those people or things that you wish

to sampleto sample• Census Census A survey of an A survey of an entire populationentire population• SampleSample A small group of a populationA small group of a population• Parameters Parameters Facts about an entire population Facts about an entire population gained from a censusgained from a census

(Notation: mean ‘(Notation: mean ‘μμ’ or standard deviation ‘’ or standard deviation ‘σσ’)’)• StatisticsStatistics Estimates of population parametersEstimates of population parameters calculated calculated

from a from a samplesample(Notation: mean ‘(Notation: mean ‘χχ’ or standard ’ or standard

deviation ‘s’)deviation ‘s’)• Representative Representative A sample that appears to A sample that appears to represent all elements of the represent all elements of the

in the correct proportionsin the correct proportions population population• BiasBias A sampling method that A sampling method that does not give every element of does not give every element of

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

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

average difference between average difference between the data values and the the data values and the mean.mean.

• This measure of spread This measure of spread applies to the mean.applies to the mean.

Advantages

Easy to calculate for large samples on calculator.

Accurate

Very useful for certain types of data

Disadvantages

Affected by outliers

Possibly not so well understood

Use Calculator to Calculate Use table to calculate

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

lower quartiles.lower quartiles.2.2. Upper quartile minus lower Upper quartile minus lower

quartile.quartile.3.3. This measure of spread This measure of spread

applies to the medianapplies to the median

Advantages

Well understood

Unaffected by outliers

Disadvantages


1.1. Find the highest and lowest Find the highest and lowest value.value.

2.2. Highest value minus the Highest value minus the lowest value.lowest value.

3.3. This measure of spread This measure of spread applies to all measures of applies to all measures of centre.centre.

RangeRange

Advantages

Well understood

Unaffected by outliers

Disadvantages


Standard Deviation by TableStandard Deviation by Table

χ χ χ – χ (χ – χ)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, doesn’t change Data values minus the Mean

Square of each of the values to the left

Final Standard Deviation is the square root of this value so s = 10

Use Calculator to Calculate

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 the Find the central value of the remaining two halves.remaining two halves.

5.5. These 2 numbers are the These 2 numbers are the upper and lower quartilesupper and lower quartiles

Calculating QuartilesCalculating Quartiles

Example (Heights of WBHS students)Example (Heights of WBHS students)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 182so median is 181so median is 181

3.3. Discard 181 and calculate middle of Discard 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 187 187

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.


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, “If I were to repeat this sampling process, would I get the same results?”


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.


Would I get the same or similar results if I repeated the same process?

• Are there outliers or extreme values that may affect the sample statistics? If so then I probably wouldn’t 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.


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!

Practice TasksPractice Tasks

Real Estate Stats

On Your CalculatorOn Your Calculator

In this part of the In this part of the presentation you can presentation you can check on exactly how check on exactly how to use your calculator to use your calculator effectively to help with effectively to help with StatisticsStatistics

Generating Random Numbers

Entering Data

Calculating Statistics

Note: For more detailed instructions on any of the example click on the step you misunderstand

Entering Data on a CalculatorEntering Data on a Calculator


Casio

FX-82

Casio

Graphic

Texas

Statistics on a CalculatorStatistics on a Calculator


Casio

FX-82

Casio

Graphic

Texas

1.1. In Stat ModeIn Stat Mode

2.2. In In list 1list 1 enter all data values enter all data values

3.3. In In list 2list 2 enter their enter 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 χχ is mean is mean, , χσχσn is std. devn is std. dev.).)

F2F1 F6

1Var XList :List11Var Freq :List22Var XList :List32Var YList :List42Var Freq :List5

EXIT

Statistics on a Statistics on a Casio Graphics CalculatorCasio Graphics Calculator

S.D. using table

Entering Data on Entering Data on Casio Graphics CalculatorCasio Graphics Calculator

Enter each data value in 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 don’t 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 List412345

EXE

1.1. Put your calculator into Put your calculator into statistics mode statistics mode

• Mode 2Mode 2

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

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

4.4. Calculate desired statisticsCalculate desired statistics• Shift 2Shift 2

1.1. χχ mean mean

2.2. χσχσnn standard standard deviationdeviation

Statistics on a Statistics on a Casio FX 82 CalculatorCasio FX 82 Calculator

Scl mode clr all

1 2 3

shift mode

M+

Shown on Screen

S.D. using table

Entering Data on Entering Data on 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 the data values as you cannot review them later to make sure that they are correct.

n =

1

M+

Statistics on a Statistics on a Texas CalculatorTexas Calculator

1.1. Put your calculator into Put your calculator into statistics modestatistics mode

1.1. 22ndnd Function DATA Function DATA

2.2. 1 - VAR1 - VAR

2.2. Enter the data carefullyEnter the data carefully 1.1. DATADATA

3.3. Calculate desired statisticsCalculate desired statistics1.1. STATVARSTATVAR

2.2. Shift between statistics with arrow Shift between statistics with arrow keyskeys1.1. nn number of number of

data valuesdata values

2.2. χχ mean mean

3.3. σχσχ standard standard deviationdeviationS.D. using table

2nd

DATA

n x Sx σx

STATVAR

Entering Data on a Entering Data on a Texas CalculatorTexas Calculator

2nd

DATA

X1 = 180

Press the Data Key to begin

Begin entering data.X1 is the data valueFollowed by the down arrow

Freq1 is that data values frequencyFollowed by the down arrowX2 is next then Freq2To check data use up arrow

Generate a Random NumberGenerate a Random Number



3.3. Choose your calculatorChoose your calculator

Casio

FX-82

Casio

GraphicTexas

Random Number on a Casio Random Number on a Casio Graphics CalculatorGraphics Calculator



3.3. In In Run Run ModeModeIntgIntg OPTN – F6 – F4 – F5 OPTN – F6 – F4 – F5

Ran#Ran# OPTN – F6 – F3 – F4 OPTN – F6 – F3 – F4

On ScreenOn Screen

Intg(529 Intg(529 × Ran# + 1)× Ran# + 1)

Population size or Strata size Starting Value

OPTN

F3 F4 F6

( )

7 8

5 ×

+1

Intg(529 × Ran# + 1)

Random Number on a Casio FX - 82Random Number on a Casio FX - 82



3.3. Ran# = 2Ran# = 2ndnd function function ··

4.4. On screenOn screen

Ran# × 529 + 1 =Ran# × 529 + 1 =

notenote Ignore any decimal in the Ignore any decimal in the answeranswer

Population size or strata size

Starting value

RAN#×529+1

·

shift

Random Number on a TexasRandom Number on a Texas



RANDIRANDI PRB PRB →→ RANDI RANDI

,, 22ndnd Function ) Function )

3.3. On Screen On Screen RANDI(1 , 529)RANDI(1 , 529)

End ValueStarting Value

2nd

PRB

)

RANDI(1,529)

Documents

A focus on Sampling and Sampling Methods