tuesday august 27, 2013 distributions: measures of central tendency & variability
TRANSCRIPT
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
scale
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.
μ
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
-3
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 = -34 - 5 = -1
μX - μ = deviation scores
-1
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 = -34 - 5 = -1
6 - 5 = +1
μX - μ = deviation scores
1
Computing standard deviation Computing standard deviation (population)(population)
Step 1: Compute the deviation scores: Subtract the population mean from every score in the distribution.
Our population2, 4, 6, 8
1 2 3 4 5 6 7 8 9 10
2 - 5 = -34 - 5 = -1
6 - 5 = +18 - 5 = +3
μX - μ = deviation scores
3
Notice that if you add up all of the deviations they must equal 0.
Computing standard deviation Computing standard deviation (population)(population)
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
Computing standard deviation Computing standard deviation (population)(population)
Step 3: Compute the Variance (the average of the squared deviations)
Divide by the number of individuals in the population.
variance = σ2 = SS/N
Computing standard deviation Computing standard deviation (population)(population)
Step 4: Compute the standard deviation. Take the square root of the population variance.
standard deviation = σ =
Computing standard deviation Computing standard deviation (population)(population)
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
Computing standard deviation Computing standard deviation (population)(population)
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).
Computing standard deviation Computing standard deviation (sample)(sample)
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
Computing standard deviation Computing standard deviation (sample)(sample)
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
Computing standard deviation Computing standard deviation (sample)(sample)
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 =
Computing standard deviation Computing standard deviation (sample)(sample)
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
◦ s
Properties of means and standard Properties of means and standard deviationsdeviations
Change/add/delete a given score
Mean Standard deviation
changes
changes
– Changes the total and the number of scores, this will change the mean and the standard deviation
Properties of means and standard Properties of means and standard deviationsdeviations
– All of the scores change by the same constant.
Mold
Change/add/delete a given score
Mean Standard deviation
Add/subtract a constant to each score
changes
changes
Properties of means and standard Properties of means and standard deviationsdeviations
– All of the scores change by the same constant.
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– All of the scores change by the same constant.
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– All of the scores change by the same constant.
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– All of the scores change by the same constant.
– But so does the mean
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
changes
M new
Properties of means and standard Properties of means and standard deviationsdeviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
changes
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
changes
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
changes
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
changes
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
changes
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
changes
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Change/add/delete a given score
Mean Standard deviation
changes
changes
Add/subtract a constant to each score
changes
Mold
Properties of means and standard Properties of means and standard deviationsdeviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Change/add/delete a given score
Mean Standard deviation
changes
changes
No change
changes
Add/subtract a constant to each score
Mold M new
Properties of means and standard Properties of means and standard deviationsdeviations
Change/add/delete a given score
Mean Standard deviation
Multiply/divide a constant to each score
changes
changes
No change
changes
Add/subtract a constant to each score
20 21 22 23 24
21 - 22 = -123 - 22 = +1
(-1)2
(+1)2
s =
M
Properties of means and standard Properties of means and standard deviationsdeviations
– Multiply scores by 2
Change/add/delete a given score
Mean Standard deviation
Multiply/divide a constant to each score
changes
changes
No change
changes
changes
changes
Add/subtract a constant to each score
42 - 44 = -246 - 44 = +2
(-2)2
(+2)2
s = 40 42 44 46 48
M
Sold=1.41