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Statistics Review

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Statistics Review

Levels of Measurement

One must know the nature of one’s variables in order to understand what manipulations are appropriate (and later, which statistical tests to use because they must be mathematically manipulated for statistics).

Nominal Level of Measurement

Ordinal Level of Measurement

Interval Level of Measurement“Continuous”

Ratio Level of Measurement

The special case of dichotomous variables:

A dichotomous variable can take one of two values.

For example:Sex:0=Male, 1=Female

Race:0=Other, 1=Hispanic

Cars:0=Other, 1=SUV

Are dichotomous variables nominal, ordinal, interval, or ratio?

- Descriptive Statistics are Used by Researchers to Report on Populations and Samples
- In Sociology:
Summary descriptions of measurements (variables) taken about a group of people

- By Summarizing Information, Descriptive Statistics Speed Up and Simplify Comprehension of a Group’s Characteristics

Summarizing Data:

- Central Tendency (or Groups’ “Middle Values” on a Variable)
- Mean
- Median
- Mode

- Variation (or Summary of Differences Within Groups on a Variable)
- Variance
- Standard Deviation

Most commonly called the “average.”

Add up the values for each case and divide by the total number of cases.

Y-bar = (Y1 + Y2 + . . . + Yn)

n

Y-bar = Σ Yi

n

What’s up with all those symbols, man?

Y-bar = (Y1 + Y2 + . . . + Yn)

n

Y-bar = Σ Yi

n

Some Symbolic Conventions in this Class:

- Y = your variable (could be X or Q or or even “Glitter”)
- “-bar” or line over symbol of your variable = mean of that variable
- Y1 = first case’s value on variable Y
- “. . .” = ellipsis = continue sequentially
- Yn = last case’s value on variable Y
- n = number of cases in your sample
- Σ = Greek letter “sigma” = sum or add up what follows
- i = a typical case or each case in the sample (1 through n)

The mean is the “balance point.”

Each IQ unit away from the mean is like 1 pound placed that far away on a scale. If IQ mean equals 110:

93

106

131

110

17 units

21 units

4 units

0 units

The scale is balanced because…

17 + 4 = 21

- Means can be badly affected by outliers (data points with extreme values unlike the rest)
- Outliers can make the mean a bad measure of central tendency or common experience

Income in the U.S.

Bill Gates

All of Us

Outlier

Mean

The middle value when a variable’s values are ranked in order; the point that divides a distribution into two equal halves.

When data are listed in order, the median is the point at which 50% of the cases are above and 50% below it.

The 50th percentile.

Class A--IQs of 13 Students

89

93

97

98

102

106

109

110

115

119

128

131140

Median = 109

(six cases above, six below)

If the first student were to drop out of Class A, there would be a new median:

89

93

97

98

102

106

109

110

115

119

128

131

140

Median = 109.5

109 + 110 = 219/2 = 109.5

(six cases above, six below)

- The median is unaffected by outliers, making it a better measure of central tendency, better describing the “typical person” than the mean when data are skewed.

Bill Gates

outlier

All of Us

- If the recorded values for a variable form a symmetric distribution, the median and mean are identical.
- In skewed data, the mean lies further toward the skew than the median.

Symmetric

Skewed

Mean

Mean

Median

Median

The most common data point is called the mode.

The combined IQ scores for Classes A & B:

80 87 89 93 93 96 97 98 102 103 105 106 109 109 109 110 111 115 119 120

127 128 131 131 140 162

BTW, It is possible to have more than one mode!

A la mode!!

- It may give you the most likely experience rather than the “typical” or “central” experience.
- In symmetric distributions, the mean, median, and mode are the same.
- In skewed data, the mean and median lie further toward the skew than the mode.

Symmetric

Skewed

Mean

Median

Mode

Mode

Median

Mean

Summarizing Data:

- Central Tendency (or Groups’ “Middle Values”)
- Mean
- Median
- Mode

- Variation (or Summary of Differences Within Groups)
- Variance
- Standard Deviation

- Often we want to know how “spread out” cases of a variable are.
- If you find the “average deviation” from the mean, you can get an idea about “spread.”
- Variance and Standard Deviation help us measure “spread.”
- Variance is a number you get in the process of calculating standard deviation.

Variance and Standard Deviation (sd) are measures of the spread of the recorded values on a variable, measures of dispersion.

The larger the variance and sd, the further the individual cases are from the mean.

The smaller the variance and sd, the closer the individual scores are to the mean.

Mean

Mean

Variance and sd are numbers that at first seems complex to calculate. We create Variance first.

Calculating variance starts with a “deviation.”

A deviation is the distance away from the mean of a case’s score.

Yi – Y-bar

If the average person’s car costs $20,000, my deviation from the mean is - $14,000!

6K - 20K = -14K

The deviation of 102 from 110.54 is?Deviation of 115?

102 - 110.54 = -8.54115 - 110.54 = 4.46

Class A--IQs of 13 Students

102115

128109

13189

98106

140119

9397

110

Y-barA= 110.54

- Since we are looking for an “average spread,” we want to add these to get total deviations:
- But if we were to do that, we would get zero every time. Why?
- We need a way to eliminate negative signs.
Squaring the deviations will eliminate negative signs...

A Deviation Squared: (Yi – Y-bar)2

Back to the IQ example,

A deviation squared for 102 is: of 115:

(102 - 110.54)2 = (-8.54)2 = 72.93(115 - 110.54)2 = (4.46)2 = 19.89

If you were to add all the squared deviations together, you’d get what we call the

“Sum of Squares.”

Sum of Squares (SS) = Σ (Yi – Y-bar)2

SS = (Y1 – Y-bar)2 + (Y2 – Y-bar)2 + . . . + (Yn – Y-bar)2

Class A, sum of squares:

(102 – 110.54)2 + (115 – 110.54)2 +

(126 – 110.54)2 + (109 – 110.54)2 +

(131 – 110.54)2 + (89 – 110.54)2 +

(98 – 110.54)2 + (106 – 110.54)2 +

(140 – 110.54)2 + (119 – 110.54)2 +

(93 – 110.54)2 + (97 – 110.54)2 +

(110 – 110.54)= SS = 2825.39

Class A--IQs of 13 Students

102115

128109

13189

98106

140119

9397

110

Y-bar = 110.54

The last step…

Looking for “average spread,” now that we’ve added, we want to divide by the number of cases.

The approximate average sum of squares is the variance.

SS/N = Variance for a population.

SS/n-1 = Variance for a sample.

Variance = Σ(Yi – Y-bar)2 / n – 1

For Class A, Variance = 2825.39 / n - 1

= 2825.39 / 12 = 235.45

How helpful is that???

To convert variance into something of meaning, let’s create standard deviation. Recall that we squared our deviations before. Let’s “unsquare” and get back to original units of IQ.

The square root of the variance reveals the average deviation of the observations from the mean.

s.d. = Σ(Yi – Y-bar)2

n - 1

For Class A, the standard deviation for IQ is:

235.45 = 15.34

The average of persons’ deviation from the mean IQ of 110.54 is 15.34 IQ points.

Review:

1. Deviation: Yi – Y-bar

2. Deviation squared: (Yi – Y-bar)2

3. Sum of squares: Σ (Yi – Y-bar)2

4. Variance: Σ(Yi – Y-bar)2 / n – 1

5. Standard deviation: Σ(Yi – Y-bar)2 / n – 1

- Larger s.d. = greater amounts of variation around the mean.
For example:

19 25 3113 25 37

Y = 25Y = 25

s.d. = 3s.d. = 6

- s.d. = 0 only when all values are the same (only when you have a constant and not a “variable”)
- If you were to “rescale” a variable, the s.d. would change by the same magnitude—if we changed units above so the mean equaled 250, the s.d. on the left would be 30, and on the right, 60
- Like the mean, the s.d. will be inflated by an outlier case value.

Many naturally occurring variables have bell-shaped distributions. That is, their histograms take a symmetrical and unimodal shape.

When this is true, you can be sure that the empirical rule will hold.

Empirical rule: If the histogram of data is approximately bell-shaped, then:

- About 68% of the cases fall between Y-bar – s.d. and Y-bar + s.d.
- About 95% of the data fall between Y-bar – 2s.d. and Y-bar + 2s.d.
- All or nearly all the data fall between Y-bar – 3s.d. and Y-bar + 3s.d.

Empirical rule: If the histogram of data is approximately bell-shaped, then:

- About 68% of the cases fall between Y-bar – s.d. and Y-bar + s.d.
- About 95% of the cases fall between Y-bar – 2s.d. and Y-bar + 2s.d.
- All or nearly all the cases fall between Y-bar – 3s.d. and Y-bar + 3s.d.

Body Pile: 100% of Cases

s.d.

15

15

15

s.d.

15

M = 100

s.d. = 15

55

70

85

115

130

145

+ or – 1 s.d.

+ or – 2 s.d.

+ or – 3 s.d.

The Normal Probability Distribution

No matter what the actual s.d. () value is, the

proportion of cases under the curve that corresponds

with the mean ()+/- 1s.d. is the same (68%).

The same is true of mean+/- 2s.d. (95%)

And mean +/- 3s.d. (almost all cases)

Because of the equivalence of all

Normal Distributions, these are often

described in terms of the Standard Normal Curve

where mean = 0 and s.d. = 1 and is called “z”

Z = # of standard deviations away from the mean

68%

68%

Z = -3 -2 -1 0 1 2 3

Z=-3 -2 -1 0 1 2 3

Converting to z-scores

To compare different normal curves, it is helpful to know how to convert data values into z-scores.

It is like have two rulers beneath each normal curve. One for data values, the second for z-scores.

IQ = 100 = 15

Values 55 70 85 100 115 130 145

Z-scores -3 -2 -1 0 1 2 3

Converting to z-scores

Z = Y –

Z = 100 – 100 / 15 = 0

Z = 145 – 100 / 15 = 45/15 = 3

Z = 70 – 100 / 15 = -30/15 = -2

Z = 105 – 100 / 15 = 5/15 = .33

IQ = 100 = 15

Values 55 70 85 100 115 130 145

Z-scores -3 -2 -1 0 1 2 3

Remember how a sampling distribution of means is created?

Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this.

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this.

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this.

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

The sample means would stack up in a normal curve. A normal sampling distribution.

z -3 -2 -1 0 1 2 3

Self-esteem 15 20 25 30 35 40

Remember how a sampling distribution of means is created?

The sample means would stack up in a normal curve. A normal sampling distribution.

2.5%

2.5%

z -3 -2 -1 0 1 2 3

Self-esteem 15 20 25 30 35 40

The sample size affects the sampling distribution:

Standard error = population standard deviation / square root of sample size

Y-bar= /n

And if we increase our sample size (n)…

Our repeated sample means will be closer to the true mean:

2.5%

2.5%

Z-3 -2 -1 0 1 2 3

z -3 -2 -1 0 1 2 3

Means will be closer to the true mean,

and our standard error of the sampling distribution is smaller:

2.5%

2.5%

Z-3 -2 -1 0 1 2 3

z -3 -2 -1 0 1 2 3

The range of particular middle percentages gets smaller:

Self-esteem 15 20 25 30 35 40

Z-3 -2 -1 0 1 2 3

95% Range

z -3 -2 -1 0 1 2 3

…But we can say that 95% of the sample means in repeated sampling will always be in the range marked by -2 over to +2 standard errors.

Self-esteem 15 20 25 30 35 40

Z-3 -2 -1 0 1 2 3

95% Range

z -3 -2 -1 0 1 2 3

Ooops!

Technically speaking, on a normal curve, 95% of cases always fall between +/- 1.96 standard deviations rather than 2.

See comparison on next slide…

Empirical Rule vs. Actuality

68%1z68%0.99z

95%2z95%1.96z

99% 2.58z

Almost all3z 99.9973%3z

…But we can say that 95% of the sample means in repeated sampling will always be in the range marked by -1.96 over to +1.96 standard errors.

Self-esteem 15 20 25 30 35 40

1.96

Z-3 -2 -1 0 1 2 3

-1.96

95% Range

z -3 -2 -1 0 1 2 3

1.96z

The sampling distribution’s standard error is a measuring stick that we can use to indicate the range of a specified middle percentage of sample means in repeated sampling.

95%

1z

68%

3z

99.99%

25

-3 -1.96 -1 0 1 1.96 3

68%

95%

99.99%

- We use that measuring stick for two things:
- Confidence Interval
- Significance Test

1.96z

95%

1z

68%

3z

99.99%

-3 -1.96 -1 0 1 1.96 3

68%

95%

99.99%

But First!

How many samples do we normally get to take?

Sample

Population

If we have only one sample to work with, how do we know anything about the sampling distribution????

25

-3 -1.96 -1 0 1 1.96 3

Like Magic, we in fact we use our sample’s standard deviation as an estimate of the population’s.

Standard error = estimate of population standard deviation / square root of sample size

Y-bar= /n

Remember why we used:

Σ(Yi – Y-bar)2 / n – 1

But what would happen if we got a sample whose mean did not equal the population mean?

25

-3 -1.96 -1 0 1 1.96 3

????

We’d center our sampling distribution over a biased estimate of the population mean!!!

25

-3 -1.96 -1 0 1 1.96 3

????

And where we drew our “middle 95%” would change!

25

95% of Sample Means

But Notice:

95% of Samples’ “95% Ranges” would contain the population mean!

25

95% of Sample Means

We will not know the true population mean, but 95% of the time the 95% range generated by your estimate of the sampling distribution will contain the true population mean!

We call this 95% Range:

95% Confidence Interval

Self-esteem 15 20 25 30 35 40

95% Ranges for different samples.

If we want that range to contain the true population mean 99% of the time (99% confidence interval) we just construct a wider interval.

Self-esteem 15 20 25 30 35 40

99% Ranges for different samples.

- We use that measuring stick for two things:
- Confidence Interval
- Significance Test

1.96z

95%

1z

68%

3z

99.99%

-3 -1.96 -1 0 1 1.96 3

68%

95%

99.99%

Confidence Interval Example:

I collected a sample of 2,500 with an average self-esteem score of 28 with a standard deviation of 8.

What if we want a 99% confidence interval? CI = Mean +/- z * s.e.

- Find the standard error of the sampling distribution:
s.d. / n = 8/50 = 0.16

- Build the width of the Interval. 99% corresponds with a z of 2.58.
2.58 * 0.16 = 0.41

- Insert the mean to build the interval:
99% C.I. = 28 +/- 0.41

The interval:27.59 to 28.41

We are 99% confident that the population mean falls between these values.

And if we wanted a 95% Confidence Interval instead?

I collected a sample of 2,500 with an average self-esteem score of 28 with a standard deviation of 8.

What if we want a 99% confidence interval? CI = Mean +/- z * s.e.

- Find the standard error of the sampling distribution:
s.d. / n = 8/50 = 0.16

- Build the width of the Interval. 99% corresponds with a z of 2.58.
2.58 * 0.16 = 0.41

- Insert the mean to build the interval:
99% C.I. = 28 +/- 0.41

The interval:27.59 to 28.41

We are 99% confident that the population mean falls between these values.

95%

X

95%

1.96

X

X

X

X

0.31

1.96

X

X

0.31

95%

X

X

27.69 to 28.31

X

95%

By centering my sampling distribution’s +/- 1.96z range around my sample’s mean...

- I can identify a range that, if my sample is one of the middle 95%, it would contain the population’s mean.
Or

- I have a 95% chance that the population’s mean is somewhere in that range.

X

By centering my sampling distribution’s +/- 1.96z range around my sample’s mean...

- I can identify a range that, if my sample is one of the middle 95%, would contain the population’s mean.
Or

- I have a 95% chance that the population’s mean is somewhere in that range.

2.58z

X

99%

99%

X

Besides construct a confidence interval, we can also do a significance test.

Let’s build a sampling distribution around our guess, 20: sample of size 100; s.d. = 10.

Sample, Y-bar

s.e. = 10/100 =

10/10 = 1

Self-doubt 16 18 20 22 24 26 28

Z: -3 -2 -1 0 1 2 3 4 5

Our sample appears to be larger than a critical value of 1.96 (outer 5% of samples) or even 2.58 (outer 1% of samples).

Sample, Y-bar

s.e. = 10/100 =

10/10 = 1

Self-doubt 16 18 20 22 24 26 28

Z: -3 -2 -1 0 1 2 3 4 5

How many z’s is our sample mean away from our guess?

Z = Y-bar – / s.e.

Z = 25 – 20 / 1

z = 5

s.e. = 10/100 =

10/10 = 1

Sample, Y-bar

Self-doubt 16 18 20 22 24 26 28

Z: -3 -2 -1 0 1 2 3 4 5

Indeed, our sample z-score is 5, well above 1.96 or 2.58. Reject the guess!

Looking in a z-Table…

Our sample has a

.0000287 % chance of

having come from a

population whose

mean is 20!

s.e. = 10/100 =

10/10 = 1

Sample, Y-bar

Self-doubt 16 18 20 22 24 26 28

Z: -3 -2 -1 0 1 2 3 4 5

- A mean is not the only statistic you can do this with.
- Other statistics, such as the difference between two groups’ means and other new statistics you will learn about, can have significance tests conducted on them.
- Just use the handy 7-step significance test format

Seven-Step Significance Test Format for Normally Distributed Sampling Distributions:

By slapping the sampling distribution for a statistic (sample’s measure of a variable) over a guess of about what the population parameter (census measure of a variable) equals, Ho, we can find out whether our sample could have been drawn from a population whose parameter equals our guess.

- Set -level for one or two tails (usually = .05)
- Find Critical Z (usually = +/- 1.96)
- Determine the null and alternative hypotheses:
Ho: Population parameter = some value (usually parameter = 0)

Ha: Population parameter some value (usually parameter 0)

- Collect Data
- Calculate Z: z = Statistic – Parameter
standard error

- Make decision about the null hypothesis
- Find P-value

Congratulations! You are now statistically sophisticated.