Tuesday August 27, 2013 Distributions: Measures of Central Tendency & Variability.

Tuesday August 27, 2013Tuesday August 27, 2013

Distributions:Measures of Central

Tendency & Variability

Today: Finish up Frequency & Today: Finish up Frequency & Distributions, then Turn to Means Distributions, then Turn to Means and Standard Deviationsand Standard Deviations

First, hand in your homework.

Any questions from last time?

Grouped Frequency TableGrouped Frequency Table

A frequency table that uses intervals (range of values) instead of single values

Pairs of Shoes

X Values Freq % Cumulative %↑ Cumulative %↓ 0-4 3 13 12 100

5-9 6 25 38 8710-14 7 29 67 6215-19 2 8 75 3320-24 4 17 92 2525-29 1 4 96 830-34 1 4 100 4Total 24 100

Frequency GraphsFrequency Graphs

Histogram Plot the

different values against the frequency of each value

Frequency GraphsFrequency Graphs

Histogram (create one for class height) Step 1: make a frequency distribution

table (may use grouped frequency tables) Step 2: put the values along the bottom,

left to right, lowest to highest Step 3: make a scale of frequencies along

left edge Step 4: make a bar above each value

with a height for the frequency of that value

Frequency GraphsFrequency Graphs Frequency polygon - essentially the

same, but uses lines instead of bars

Properties of distributionsProperties of distributions

Distributions are typically summarized with three features

Shape Center Variability (Spread)

Shapes of Frequency Shapes of Frequency DistributionsDistributions

Unimodal, bimodal, and rectangular

Shapes of Frequency Shapes of Frequency DistributionsDistributions

Symmetrical and skewed distributions

Normal and kurtotic distributions

Next TopicNext TopicIn addition to using tables and graphs to describe distributions, we also can provide numerical summaries

Chapters 3 & 4Chapters 3 & 4Measures of Central Tendency

◦Mean◦Median◦Mode

Measures of Variability◦Standard Deviation & Variance

(Population)◦Standard Deviation & Variance (Samples)

Effects of linear transformations on mean and standard deviation

Self-Monitor you Understanding

These topics should all be review from PSY 138, so I will move fairly quickly through the lecture.

I will stop periodically to ask for questions.

Please ask if you don’t understand something!!!

If you are confused by this material, it will be very hard for you to follow and keep up with later topics.

Describing distributionsDescribing distributionsDistributions are typically described

with three properties:◦ Shape: unimodal, symmetrical, skewed,

etc.◦ Center: mean, median, mode◦ Spread (variability): standard deviation,

variance

Describing distributionsDescribing distributionsDistributions are typically described

with three properties:◦ Shape: unimodal, symmetric, skewed, etc.◦ Center: mean, median, mode◦ Spread (variability): standard deviation,

variance

Which center when?Which center when?Depends on a number of factors, like

scale of measurement and shape.◦ The mean is the most preferred measure

and it is closely related to measures of variability

◦ However, there are times when the mean isn’t the appropriate measure.

Which center when?Which center when?

Use the median if: The distribution is skewed The distribution is ‘open-ended’

(e.g. your top answer on your questionnaire is ‘5 or more’)

Data are on an ordinal scale (rankings)Use the mode if the data are on a nominal

Self-monitor your understandingWe are about to turn to a

discussion of calculating means.Before we move on, any

questions about when to use which measure of central tendency?

The MeanThe Mean The most commonly used measure of center The arithmetic average

◦ Computing the mean

– The formula for the population mean is (a parameter):

– The formula for the sample mean is (a statistic):

Add up all of the X’s

Divide by the total number in the population

Divide by the total number in the sample

• Note: Sometimes ‘ ’ is used in place of M to denote the mean in formulas

The MeanThe MeanNumber of shoes:

2,2,2,5,5,5,7,86,10,10,12,12,13,14,14,15,15,20,20,20,20,25,30

• Suppose we want the mean of the entire group? Can we simply add the two means together and divide by 2?

• (4.5 + 16)/2 = 20.5/2 = 10.25• NO. Why not?

= (2+2+2+5+5+5+7+8)/8 = 36/8 = 4.5

= (6+10+10+12+12+13+14+14+15+15+20+20+20+20+ 25+30)/16 = 256/16 = 16

The Weighted MeanThe Weighted MeanNumber of shoes:

2,2,2,5,5,5,7,8,6,10,10,12,12,13,14,14,15,15,20,20,20,20,25,30

Mean for men = 4.5 Mean for women = 16

= [(4.5*8)+(16*16)]/(8+16) =(36+256)/24)

= 292/24 = 12.17

Need to take into account the number of scores in each mean ( & )

The Weighted MeanThe Weighted Mean

Number of shoes: 2,2,2,5,5,5,7,8,6,10,10,12,12,13,14,14,15,15,20,20,20,20,25,30

• Both ways give the same answer

Let’s check:

= [(4.5*8)+(16*16)]/(8+16) = (36+256)/24 = 292/24 = 12.17

= 256/24=12.17

Self-monitor your understandingWe are about to move on to a

quick discussion of calculating the median and mode.

Before we move on, any questions about the formulae for the population mean, sample mean?

Questions about the weighted mean?

The medianThe median

The median is the score that divides a distribution exactly in half. Exactly 50% of the individuals in a distribution have scores at or below the median.◦ Case1: Odd number of scores in the distributionStep1: put the

scores in order Step2: find the middle score

Step1: put the scores in order Step2: find the middle two scores Step3: find the arithmetic average of the two middle scores

– Case2: Even number of scores in the distribution

The modeThe mode

The mode is the score or category that has the greatest frequency. ◦ So look at your frequency table or graph and

pick the variable that has the highest frequency.

so the mode is 5 so the modes are 2 and 8

Note: if one were bigger than the other it would be called the major mode and the other would be the minor mode

major modeminor mode

Self-monitor your understandingWe are about to switch to the

topic of measures of variabilityBefore we move on, any

questions about measures of central tendency?

Describing distributionsDescribing distributionsDistributions are typically described with three properties:◦ Shape: unimodal, symmetric, skewed, etc.◦ Center: mean, median, mode◦ Spread (variability): standard deviation,

variance

Variability of a distributionVariability of a distribution

Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.◦ In other words variabilility refers to the degree of

“differentness” of the scores in the distribution.

High variability means that the scores differ by a lot

Low variability means that the scores are all similar

Standard deviationStandard deviation

The standard deviation is the most commonly used measure of variability.◦ The standard deviation measures how far off all

of the scores in the distribution are from the mean of the distribution.

◦ Essentially, the average of the deviations.

Computing standard deviation Computing standard deviation (population)(population)

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution.

Our population2, 4, 6, 8

1 2 3 4 5 6 7 8 9 10

2 - 5 = -3

μX - μ = deviation scores

1 2 3 4 5 6 7 8 9 10

2 - 5 = -34 - 5 = -1

1 2 3 4 5 6 7 8 9 10

2 - 5 = -34 - 5 = -1

6 - 5 = +1

Step 1: Compute the deviation scores: Subtract the population mean from every score in the distribution.

1 2 3 4 5 6 7 8 9 10

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

Notice that if you add up all of the deviations they must equal 0.

Step 2: Get rid of the negative signs. Square the deviations and add them together to compute the sum of the squared deviations (SS).

SS = Σ (X - μ)2

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

X - μ = deviation scores

= (-3)2+ (-1)2+ (+1)2+ (+3)2

= 9 + 1 + 1 + 9 = 20

Step 3: Compute the Variance (the average of the squared deviations)

Divide by the number of individuals in the population.

variance = σ2 = SS/N

Step 4: Compute the standard deviation. Take the square root of the population variance.

standard deviation = σ =

To review:◦ Step 1: compute deviation scores◦ Step 2: compute the SS

SS = Σ (X - μ)2

◦ Step 3: determine the variance take the average of the squared deviations divide the SS by the N

◦ Step 4: determine the standard deviation take the square root of the variance

Self-monitor your understandingWe are about to learn how to calculate

sample standard deviations.Before we move on, any questions about

how to calculate population standard deviations?

Any questions about these terms: deviation scores, squared deviations, sum of squares, variance, standard deviation?

• Any questions about these symbols: SS

The basic procedure is the same.◦ Step 1: compute deviation scores◦ Step 2: compute the SS◦ Step 3: determine the variance

This step is different

◦ Step 4: determine the standard deviation

Computing standard deviation Computing standard deviation ((samplesample))

Computing standard deviation Computing standard deviation (sample)(sample)

Step 1: Compute the deviation scores◦ subtract the sample mean from every individual in our

distribution.

Our sample2, 4, 6, 8

1 2 3 4 5 6 7 8 9 10

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

X - M = Deviation Score M

Step 2: Determine the sum of the squared deviations (SS).

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

= (-3)2+ (-1)2+ (+1)2+ (+3)2

= 9 + 1 + 1 + 9 = 20

X - M = deviation scores SS = Σ(X - M)2

Apart from notational differences the procedure is the same as before

Step 3: Determine the variance

Population variance = σ2 = SS/N

Recall:

μX1 X2X3X4

The variability of the samples is typically smaller than the population’s variability

Step 3: Determine the variance

Population variance = σ2 = SS/N

Recall:

The variability of the samples is typically smaller than the population’s variability

Sample variance = s2

To correct for this we divide by (n-1) instead of just n

Step 4: Determine the standard deviation

standard deviation = s =

Self-monitor your understandingNext, we’ll find out how changing our

scores (adding, subtracting, multiplying, dividing) affects the mean and standard deviation.

Before we move on, any questions about the sample standard deviation?

About why we divide by (n-1)?About the following symbols:

◦ s2

Properties of means and standard Properties of means and standard deviationsdeviations

Change/add/delete a given score

Mean Standard deviation

changes

– Changes the total and the number of scores, this will change the mean and the standard deviation

– All of the scores change by the same constant.

Add/subtract a constant to each score

changes

– But so does the mean

changes

– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same

changes

No change

changes

Mold M new

Multiply/divide a constant to each score

changes

No change

changes

20 21 22 23 24

21 - 22 = -123 - 22 = +1

– Multiply scores by 2

Multiply/divide a constant to each score

changes

No change

changes

42 - 44 = -246 - 44 = +2

s = 40 42 44 46 48

Sold=1.41

Tuesday August 27, 2013 Distributions: Measures of Central Tendency & Variability.

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