TYPES There are several TYPES of variables that reflect characteristics of the data Ratio Interval...

There are several TYPESTYPES of variablesthat reflect characteristics of the data

RatioIntervalOrdinalNominal

Ratio scale

constant size interval between adjacent values on the measurement scale

existence of a meaningful zero point

Interval scale

constant size interval between adjacent values on the measurement scale no true zero value

N

S

EW 0-10

10

Ordinal scale

data that convey only relative magnitude

Tall Medium Short

Dark

Medium

Light

Nominal scale

data in which there is no meaningful numerical information

SingleMarriedDivorcedWidowed

Another useful classification

Continuous

Discrete

data can take-on any value

data can take-on only certain values

Eg height 150 to 210cm rangeBill - 174.25 cmBill - 174.25 cm

Eg # of hands 0 to 3 rangeBill - 2 handsBill - 2 hands

2 more important issues with data

AccuracyAccuracy how close is a measured value to the real value

PrecisionPrecision how close repeated measurements are to one another

Let’s say Bill’s real real heightis 174.25 cm.

AccuratePrecise

174.25

174.25

174.25

174.25

174.25

174.25

Not AccurateNot Precise

172

178

171

174

182

168

Not AccuratePrecise

170.25

170.25

170.25

170.25

170.25

170.25

Frequency DistributionFrequency Distribution

occurrence of the various values observed for the variable

raw frequency counts

relative frequency counts divided by total number of observations

Name Height (cm) Hair Colour

Anne 168 Brown

Rishi 178 Black

Bill 183 Brown

Cristin 172 Brown

Rich 175 Black

Variable: Hair ColourVariable: Hair Colour

Sample size = 5

Frequency of Black Hair = 2Frequency of Brown Hair = 3

Must add to 5

Relative Frequency of Black Hair = 2/5 = 0.4Relative Frequency of Brown Hair = 3/5 = 0.6

Must add to 1

Variable: HeightVariable: Height

Sample size = 5

Frequency of 168 cm = 1Frequency of 172 cm = 1Frequency of 175 cm = 1Frequency of 178 cm = 1Frequency of 183 cm = 1

Relative Frequency of 168 cm = 1/5 = 0.2Relative Frequency of 172 cm = 1/5 = 0.2Relative Frequency of 175 cm = 1/5 = 0.2Relative Frequency of 178 cm = 1/5 = 0.2Relative Frequency of 183 cm = 1/5 = 0.2

Make categories

Eg. Number above and number below mid-point of range

Range: Maximum - Minimum

183 cm - 168 cm = 15 cm

Mid-point: half way between Min and Max

= Min + (Range / 2)= 168 cm + 7.5 cm= 175.5 cm

Frequency of Heights Below 175.5 cm = 3Frequency of Heights Above 175.5 cm = 2

Relative Frequency of Heights Below 175.5 cm = 3/5 = 0.6Relative Frequency of Heights Above 175.5 cm = 2/5 = 0.4

Could make THREETHREE categories

Divide range by 3: 15 cm / 3 = 5 cm

Category 1: 168 cm to 168 cm + 5 cm 168 cm to 173 cm



Frequency of Heights in 168 cm to 172 cm = 2Frequency of Heights in 173 cm to 178 cm = 2Frequency of Heights in 179 cm to 184 cm = 1

Relative Frequency of Heights in 168 cm to 172 cm = 2/5 = 0.4



19 252333 255120 255721 259418 260021 262222 263717 263729 266326 266519 272219 273322 275030 275018 276918 276915 277825 278220 280728 282132 283531 283536 283628 286325 287728 287717 290629 292026 292017 292017 292024 294835 2948

25 297725 297729 297719 297727 299231 300533 303321 304219 306223 306221 306218 307618 307632 308019 309024 309022 309022 310023 310422 313230 314719 317516 317521 320330 320320 320317 322517 322523 323224 323228 323426 326020 3274

24 327428 330320 331722 331722 331731 332123 333116 337416 337418 340225 341632 343020 344423 345922 346032 347330 347520 348723 354417 357219 357223 358636 360022 361424 361421 362919 362925 363716 364329 365129 365119 365119 3651

30 369924 372819 375624 377023 377020 377025 379030 379922 382718 385616 386032 386018 388429 388433 391220 394028 394114 394128 396925 398316 399720 399726 405421 405422 411125 415331 416735 417419 423824 459345 499028 70929 1021

34 113525 133025 147427 158823 158824 170124 172921 179032 181819 188525 189316 189925 192820 192821 192824 193621 197020 205525 205519 208219 208426 208424 210017 212520 212622 218727 218720 221117 222525 224020 224018 228218 2296

20 229621 230126 232531 235315 235323 236720 238124 238115 238123 239530 241022 241017 241423 242417 243826 244220 245026 246614 246628 246614 249523 249517 249521 2495

Mother’s age and babies birth weight data from Massachusetts

Range of the Birth Weight data: Minimum: 709 g Maximum: 4990 gDifference: 4281 g

Let’s say we want to look at the distribution of data across 10 categories.

Each category would span 428.1 g, but for convenience we’ll round to 430 g.

Also, instead of starting our first category at 709 g we’ll use 700g

Category12345678910

Range700-11301131-15601561-19901991-24202421-28502851-32803281-37103711-41404141-47504751-5000

0.0158730160.0158730160.0740740740.1534391530.179894180.2328042330.1746031750.1216931220.0211640210.010582011

3314293444332342

Freq. Rel. Freq.

Previous breakdown ok as long as I have measured weight to the nearest gram.

BUT, if I’ve measure to the nearest 0.1 gram

--> my categories may miss some observations

So need to adjust…

Category123456789

10

Range700-1130

1131-15601561-19901991-24202421-28502851-32803281-37103711-41404141-47504751-5000

Range700-1130.9

1131-1560.91561-1990.91991-2420.92421-2850.92851-3280.93281-3710.93711-4140.94141-4750.94751-5000 .9

Measured to the nearest gram Measured to the nearest 0.1 gram

HistogramHistogram - graphical representation of a frequency distribution

0

0.5

1

1.5

2

2.5

3

Brown Hair Black Hair

Freq

uen

cy

Hair colour

0

10

20

30

40

50

1 2 3 4 5 6 7 8 9 10

Birth Weight Category

Freq

uen

cyFrequency distribution of neonatal birth weight

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 10


Rela

tive F

req

uen


Category12345678910

Range700-11301131-15601561-19901991-24202421-28502851-32803281-37103711-41404141-47504751-5000

Mid-point915134617762206263630663496392643564966

Birth Weight Category Mid-point

Freq

uen


Category123456789

10

Range700-1130

1131-15601561-19901991-24202421-28502851-32803281-37103711-41404141-47504751-5000

0.01580.01580.074070.153430.179890.232800.174600.121690.021160.01058

3314293444332342

Freq. Rel. Freq. Cum. Freq.0.01580.03170.10580.25920.43910.67190.84650.96820.98941.0

Cumulative FrequencyCumulative Frequency - Cum. Freq. at any category is equal to the frequency at that category plus the frequency in each previous category.

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10


Cu

mula

tive F

requ

en


Measures of Central TendencyMeasures of Central Tendency

MeanMedianMode

These generally tell you where the majority of the observations lie Each one tells something slightly different

Average

Middle Value

Most Frequent Value

The Mean:The Mean:

The mean is calculated by summing the observed values and dividing the sum by the total number of observations.

Population Mean = μ

Sample Mean = X

A die has 6 sides, 1 dot, 2, 3, 4, 5, and 6

dots5.36

654321

dotsX 33

432

N

XXXX N

...321

n

XXXXX n

...321

N

XN

ii

1

n

X

X

n

ii

1

RishiAnneBillCristinRich

Observationi

HeightXi

12345

172185132191205

n = 5

1775

885'

n

sXX

= 885

19 252333 255120 255721 259418 260021 262222 263717 263729 266326 266519 272219 273322 275030 275018 276918 276915 277825 278220 280728 282132 283531 283536 283628 286325 287728 287717 290629 292026 292017 292017 292024 294835 2948

25 297725 297729 297719 297727 299231 300533 303321 304219 306223 306221 306218 307618 307632 308019 309024 309022 309022 310023 310422 313230 314719 317516 317521 320330 320320 320317 322517 322523 323224 323228 323426 326020 3274

24 327428 330320 331722 331722 331731 332123 333116 337416 337418 340225 341632 343020 344423 345922 346032 347330 347520 348723 354417 357219 357223 358636 360022 361424 361421 362919 362925 363716 364329 365129 365119 365119 3651

30 369924 372819 375624 377023 377020 377025 379030 379922 382718 385616 386032 386018 388429 388433 391220 394028 394114 394128 396925 398316 399720 399726 405421 405422 411125 415331 416735 417419 423824 459345 499028 70929 1021

34 113525 133025 147427 158823 158824 170124 172921 179032 181819 188525 189316 189925 192820 192821 192824 193621 197020 205525 205519 208219 208426 208424 210017 212520 212622 218727 218720 221117 222525 224020 224018 228218 2296

20 229621 230126 232531 235315 235323 236720 238124 238115 238123 239530 241022 241017 241423 242417 243826 244220 245026 246614 246628 246614 249523 249517 249521 2495

n = 189

189

1

556540i

iX

189

1

556540i

iXn = 189

656.2944189

556540'

n

sXX

Another way to calculate the meanAnother way to calculate the mean

Suppose you had a frequency distribution for the number of cancerous moles on people who regularly visit Club Med

# cancerous moles(X)

Frequency(f)

012345

8481021

# cancerous moles(x)

Frequency(f)

012345

8481021

04163085

f * x

n = f’s

X’s = f*x

n = 33 f*x = 63

909.133

63*

f

xfX

The Mode:The Mode: the most frequently occurring value in a set of measurements

0

10

20

30

40

50

1 2 3 4 5 6 7 8 9 10


Fre

quen

cy

Frequency distribution of neonatal birth weight

Category123456789

10

Range700-1130

1131-15601561-19901991-24202421-28502851-32803281-37103711-41404141-47504751-5000

0.0158730160.0158730160.0740740740.1534391530.179894180.2328042330.1746031750.1216931220.0211640210.010582011

3314293444332342

Freq. Rel. Freq.

Mid-point is 3065.5 --> report the MODE as 3065.5

The Median: the middle measurement of a set of data

--> data must be ordered

Heights (cm)178143123189187205168173198

Ordered Heights (cm)123143168173178187189198205

Observation (X)123456789

Median is 178 cm

Heights (cm)178143123189187205168173198162

Ordered Heights (cm)123143162168173178187189198205

Observation (X)123456789

10

Middle observation is 5.5 --> median is midway between observation 5 and observation 6

Median is (173+178)/2 = 175.5

General formula for Median:

If n is an oddodd number:

2/)1( nX

2/)19( X

178)5( X

General formula for Median:

If n is an eveneven number:

2/)1( nX

2/)110( X

)5.5(X

265 XX

5.1752

178173

# cancerous moles(X)

Frequency(f)

012345

8481021

Cumulative Frequency

81220303233

M = X(n+1)/2=X17=2

000000001111

222222223333

333333445

Category123456789

10

Range700-1130

1131-15601561-19901991-24202421-28502851-32803281-37103711-41404141-47504751-5000

36204983127160183187189

3314293444332342

Freq. Cum. Freq.

M = X(n+1)/2 = X190/2 = X95

Median =

(lower limit of class) + ((0.5*n - cum.freq.)/#obs in interval)(interval size)

= 2851 + ((0.5*189- 83)/44) * (430)

= 2851 + (94.5-83)/44 *430

= 2963.4

Of the previous class

0

10

20

30

40

50

1 2 3 4 5 6 7 8 9 10


Fre

quen


05

1015202530354045

1 2 3 4 5 6 7 8 9 10 11 12 13

Symetrical, unimodal distribution

Mean, Mode and Median

02468

1012141618

1 2 3 4 5 6 7 8 9 10 11 12 13

Symetrical, bimodal distribution

MeanMedain

ModeMode

05

1015202530354045

1 2 3 4 5 6 7 8 9 10 11 12 13

Asymmetric distribution

Mode Median Mean

05

1015202530354045

1 2 3 4 5 6 7 8 9 10 11 12 13

Asymmetric distribution

Mean Median Mode

Measures of Dispersion and Variability

0

10

20

30

40

50

1 2 3 4 5 6 7 8 9 10


Fre

quen


0500

1000150020002500300035004000450050005500

0 0.2 0.4 0.6 0.8 1 1.2

Birt

h W

eigh

t (g)

Mean

Maximum

Minimum

Ran

ge

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5

0500

1000150020002500300035004000450050005500

0 0.2 0.4 0.6 0.8 1 1.2

Birt

h W

eigh

t (g)

Mean

Maximum

Minimum

Observationi

Deviation

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5

Average Deviation from the Mean

--> on average, how much do the individual observations differ from the mean?

n

XX in

i)(

1

Xi

1.21.41.61.82.02.22.4

XX i XX i 2XX i 1.2-1.8 = -0.6

-0.4-0.20.00.20.40.6

X=12.6n=7

8.17

6.12X

i1234567

07

1

XX i

i

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5

Average Absolute Deviation from the Mean

--> on average, how much do the individual observations differ from the mean?

n

XX in

i

1

Xi

1.21.41.61.82.02.22.4

XX i XX i 2XX i 1.2-1.8 = -0.6

-0.4-0.20.00.20.40.60.0X=12.6

n=78.1

7

6.12X

i1234567

|1.2-1.8| = 0.60.40.20.00.20.40.6

34.07

4.2

7

7

1

XX i

i

Sum of Squared Deviations

n

ii XXSS

1

2)(

“Sum of Squares”

Xi

1.21.41.61.82.02.22.4

XX i XX i 2XX i -0.6-0.4-0.20.00.20.40.60.0X=12.6

n=78.1

7

6.12X

i1234567

0.60.40.20.00.20.40.6

0.34

(-0.6)2 = 0.360.160.04

00.040.160.36

12.1)(1

2

n

ii XX

1.12

Variance

--> mean sum of squares

1

)(1

2

2

n

XXs

n

ii

N

Xn

ii

1

2

2

)( Population

Sample

Xi

1.21.41.61.82.02.22.4

XX i XX i 2XX i -0.6-0.4-0.20.00.20.40.60.0X=12.6

n=78.1

7

6.12X

i1234567

0.60.40.20.00.20.40.6

0.34

(-0.6)2 = 0.360.160.04

00.040.160.361.12

1867.06

12.1

1

)(1

2

2

n

XXs

n

ii

Standard Deviation

2ss

2 Population

Sample

Coefficient of Variation

X

sV

--> allows comparison of variability among samples measured in different units or scales.

S expressed as a % of the mean

0

0.5

1

1.5

2

2.5

3

Mean DeviationVarianceStandard deviationCV

0.340.18670.430.24

0.260.13670.370.21

Standard Error of the MeanStandard Error of the Mean

Recall: x and s are estimates of Recall: x and s are estimates of μμ and and σσ

How good are these measures??How good are these measures??

Need level of uncertainty (due to sampling Need level of uncertainty (due to sampling error) in the mean:error) in the mean:

SEx = s/√ n

Confidence IntervalsConfidence Intervals

SE = measure of how far x is likely to be SE = measure of how far x is likely to be from from μμ

2 * SE = 95% confidence2 * SE = 95% confidence

I.e. μ is inside 2 * SE 95% of the timeI.e. μ is inside 2 * SE 95% of the time

Reporting variability about the mean.

Text

In a table as in previous slide.

Or, for example, in a manuscript, I might write:

The mean (± 95% CI) for the random samples of 100, 50, 25 and 10 was 24.84079 (±0.1816), 24.91241(±0.31996), 24.86719 (±0.40142) and 25.16212 (±0.859) respectively. You are not restricted to using the confidence intervals when reporting variability about the mean, ie I could have used mean ± std dev, or mean ± std error

Graphically: Box Plot or Box and Whisker Plot

Type of Mother

Ne

on

ate

We

igh

t (g

)

2550

2650

2750

2850

2950

3050

3150

3250

Non-smokers Smokers

MeanStandard Error95% CI


Type of Mother

Ne

on

ate

We

igh

t (g

)

2550

2650

2750

2850

2950

3050

3150

3250

Non-smokers Smokers

Mean

95% CI


Type of Mother

Ne

on

ate

We

igh

t (g

)

0

500

1000

1500

2000

2500

3000

3500

4000

Non-smokers Smokers

Mean

95% CI

TYPES There are several TYPES of variables that reflect characteristics of the data Ratio Interval...

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Transcript of TYPES There are several TYPES of variables that reflect characteristics of the data Ratio Interval...