Module 5: Measures of Dispersion - - Get a Free Blog

Psy102: Fundamental Statistics inPsychology (ADMU)1st Semester 2009-2010

Prepared by Niel Steve M. Kintanar

Module 5:Measures of Dispersion

“Statistics is the study ofvariability”

-Fr. Jaime Bulatao, Founder,Ateneo Psychology Department

What we learned so far….

√ Summarize the data using tables andgraphs (histograms and polygons).

√ But you can also summarize the datanumerically!

MEASURES OFDISPERSION

MEASURES OFCENTRALTENDENCY

√



Your Standard Deviation is…

1. A measure of error for the mean2. A measure of dispersion and spread.3. A measure of comparison against the

whole distribution (normal curve)

But Central Measures…Ψ Yields an incomplete picture of the data andcan be misleading.

Ψ We always look at how much error we madein summarizing the data.

Ψ Another term for error is variability (orinconsistencies, unevenness).

Ψ Measures of dispersion or variability is anindex of how the scores are scattered aroundthe center of the distribution. Variability is alsoknown as spread, width, or dispersion.(Closely hug to the center? Or are they spreadaway from the center?)



Measures of Dispersions

Ψ Range

Ψ Variance

Ψ Standard Deviation

A research scenario…You were tasked to find out the use of various drugsspecifically alcohol from two colleges (School ofSocial Sciences & School of Management). Studentswere sampled and were asked how frequently theyhad gotten drunk in the previous six months. Thestudents answered.

SOSS (X1) SOM (X2)1 20 36 11 2

mean X1 = 2 mean X2 = 2

Whoconsumedmore alcoholfor the pastsix months?



1. RangeΨ A quick but a rough measure ofvariability.

Ψ Range = Highest score – lowest score + 1

Ψ Advantage: It is quick and easy.

Ψ Disadvantage: Totally dependenton only two score values (crudeindex)

1. What are the Range?SOSS (X1) SOM (X2)

1 20 36 11 2

mean X1 = 2 mean X2 = 2

Range (X1) = 6 – 0 + 1= 7

Range (X2) = 3 – 1 + 1= 3

Interpretation: The SOSS students reportedhaving been drunk within the range of none to 6(with a 7 index range) while the SOM studentsreported having been drunk within the range ofone to three (with 3 as index range.)



2 - 3. Variance & StandardDeviation

Ψ a measure of variability that overcomesthe problem of absolute values and can beused for advanced statistical procedures.

Ψ Advantage: It gives appropriatelygreater emphasis to extreme scores (moresensitive to the degree of deviation in thedistribution.

Ψ Variance is the average of thesummation of the squared deviationswhile the standard deviation is the squareroot of the variance.

Remember? The Measure of Error of the Mean

Happinessscore

Deviation score(x-mean)

Squared Deviations(x-mean)2

4 -2 45 -1 16 0 06 0 06 0 06 0 06 0 07 1 18 2 4

Σx= 54 Σ(x-mean)= 0 SS or Σ(x-mean)2=10

Ψ The SS or thesum of squares

(summation of thesquared deviation)

is the measure of error ofthe mean.

Ψ where the sum ofsquares (SS) of thedeviation are at the

minimum.(LEAST SQUARES

CRITERION)



3. Computing the Variance &Standard Deviation

Variance (S2) = --------------------- = --------Σ ( x – mean)2

n

SS

n

Standard Deviation = --------------------- = ------Σ ( x – mean)2

n

SS

nS or Sd

Definitional formula

3. Going Back to Our Alcohol Data…Get S2 and S.

SOSS (X1) (x-mean) ( x – mean )2 SOM (X2) (x-mean) ( x – mean )2

1 -1 1 2 0 00 -2 4 3 1 16 4 16 1 -1 11 -1 1 2 0 0

mean X1 = 2 Σ(x-mean)= 0 SS= 22 mean X2 = 2 Σ(x-mean)= 0 SS = 2

Variance (S2) = -------- = ------ = 5.5SS

n

22

4Variance (S2) = -------- = ------ = 0.50

SS

n

2

4

S or Sd = ------ = 5.5 = 2.35SS

nS or Sd = ------ = 0.50 = 0.71

SS

n



3. Interpreting the variance andstandard deviation.

SOSS SOMMean = 2 Mean = 2

R = 7 R = 3S2 = 5.5 S2 = 0.50

S or Sd = 2.35 S or Sd = 0.71

Interpretation: Clearly the SOMdistribution scores have lessvariability (less spread, more huggedto the center) than the SOSSdistribution scores (more spread).

SOSS

SOM

3. Questions on variance and standarddeviation.

A. Why did we square the mean deviations? (Refer tonotes.)

B. Why are we getting the square root of the variance forthe standard deviation?

Having squared the deviations, the unit of measurement isaltered, making the variance difficult to interpret. (If thedata were IQ scores, the variance would be in IQ2 units.The variance is expressed as the square of whatever unitexpresses our data.

To return to the original unit of measurement we use thestandard deviation – a square root of the variance!



Now Your Turn!SOH (X1) SOSE (X2)

4 34 56 56 7

Mean = ? Mean = ?R = ? R = ?

S2 = ? S2 = ?S or Sd = ? S or Sd = ?

You were tasked to find out theuse of various drugs specificallyalcohol from two colleges (Schoolof Humanities & School of Science& Engineering). Students weresampled and were asked howfrequently they had gotten drunkin the previous six months. Thestudents answered.

Interpret your Results!SOH (X1) SOSE (X2)

4 34 56 56 7

Mean = 5 Mean = 5R = 3 R = 5

S2 = 1 S2 = 2S or Sd = 1 S or Sd = 1.41



Computational Formula of Variance andStandard Deviation!

Standard Deviation (S) = ---------- - x 2Σ x2

n

Variance (S2) = ---------- - x 2Σ x2

n

From sample to population…

Sample Population

Mean µ

Error S or Sd σx

Central Limit Theorem (more in the next module!)1. The mean of a sampling distribution of means (the means of

means) is equal to the true population mean. (The sample mean isalready an unbiased estimate to the population mean.)

2. The standard deviation of a sampling distribution of means issmaller than the standard deviation of the population.(The use of n-1)



In Inferential Statistics…we use n-1 in the Computational Formula of Variance

and Standard Deviation!

* Unlike the mean, the sample Sd is always biased. (Fromwhat theory?) Sd is always smaller than δ. We use n-1 to

correct the error.

Standard Deviation = --------------------- = ------Σ ( x – mean)2

n - 1

SS

n - 1S or Sd

Standard Deviation (S) = Σ x 2 - (Σ x) 2

n

n - 1

The meaning of standard deviation…1. The standard deviation represents the average variability (or error)

in the distribution, because it measures the average of deviationsfrom the mean.

2. The greater the variability around the mean, the larger thestandard deviation.

3. The standard deviation is a useful device for measuring thedegree of variability in a distribution or for comparing the variabilityin different distributions. It is employed, and quite frequently, forcalibrating the relative standing (position or Z score, more in thenext module) of individual scores within a distribution.

4. THE SD IN THIS SENSE IS A STANDARD AGAINST WHICH WEASSESS THE PLACEMENT OF ONE SCORE (THE NUMBEROF TIMES YOU WERE DRUNK) WITHIN THE WHOLEDISTRIBUTION (THE NUMBER OF TIMES ALL THE SAMPLEDPARTICIPANTS GOT DRUNK)



Your Standard Deviation is…

1. A measure of error for the mean2. A measure of dispersion and spread.3. A measure of comparison against the

whole distribution (normal curve)

-3Sd -2Sd -1Sd +1Sd +2Sd +3Sd

The Normal Curve6 σ or standarddeviations includepractically all casesunder any normaldistribution

x



The Normal Curve Unless adistribution ishighly skewed,approximately two-thirds of its scoresfall within onestandard deviationabove and belowthe mean. Usuallyreferred as thenormal range.

-3Sd -2Sd -1Sd +1Sd +2Sd +3Sdx


The Normal Curve (Using SOHsample)

x2 3 4 5 6 7 8

Interpretation: Theaverage number oftimes SOH studentsget drunk in a month is5 with Sd of 1, thenthe normal range maybe defined as 4-6times a month andapproximately twothirds of the students.

Mean = 5, Sd = 1




The Normal Curve (Using SOSEsample)

x0.77 2.18 3.59 5 6.41 7.82 9.23

Interpretation: Theaverage number oftimes SOSE studentsget drunk in a month is5 with Sd of 1.41, thenthe normal range maybe defined as 3.59-6.41 times a monthand approximately twothirds of the students.

Mean = 5, Sd = 1.41

What’s Next?

The Normal Curve andStandard Scores

THANK YOU VERYMUCH!!!!

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Module 5: Measures of Dispersion - - Get a Free Blog