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Statistics Primer

ORC Staff:

Xin Xin (Cindy)

Ryan Glaman

Brett Kellerstedt

Quick Overview of Statistics

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Descriptive vs. Inferential Statistics

Descriptive Statistics: summarize and describe data (central tendency, variability, skewness)

Inferential Statistics: procedure for making inferences about population parameters using sample statistics

Sample

Population

Measures of Central Tendency

Raw data Simple frequency distribution

Group frequency distribution

Notations

Mode Pick out the value (s) occurring more than any other value.

Pick out the value (s) with the highest frequency.

= Difference between the freq. of modal class and the freq. of the next lower class.

= Difference between the freq. of modal class and the freq. of the next higher class.

L1 = Lower class boundary of the modal class

c = class width of the modal class

Median

1.Order data2.Determine

median position = (n+1)/2

3.Locate median based on step 2.

1.Order data2.Determine median

position = (n+1)/23.Locate median

based on step 2 using the freq. column

Lm=lower class boundary of median classn = sample sizeC.F. = sum of all frequencies lower than the median classfmed = frequency of the median class

c = class width of the median class

Mean Add up all the data values and divide by the number of values.

Find the product of all the values and their frequencies ; then add all the products; and finally divide by the total frequency.

Find the product of all the midpoints and their frequencies ; then add all the products; and finally divide by the total frequency.

X = the actual values (for raw data and ungrouped freq. dist.)

= midpoints (for group freq. dist.)f = frequency n = sample sizeN = population size

= summation or sum of

c

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11L

4

1

2

cf

FCnL

medm

.).2(

n

Xor

N

X

f

fX

f

fX

Description Applicability Advantage Disadvantage

Range Difference between the largest and the smallest value in the data.

1.Interval/ratio2.No outliers exist

1.Simple to calculate

1.Highly influenced by outliers.

2.Does not use all data

Mean deviation

It measures the average absolute deviations from the mean. Uncommonly used

1.Interval/ratio2.When no outliers

exist

1.Use all the data2.Easy to interpret

1.Not resistant to outliers

2.Does not yield any further useful statistical properties.

Variance/ standard deviation

Variance is the average squared deviations from the mean.

Standard deviation is square root of the variance. Commonly used.

1.Interval/ratio2.When no outliers

exist

1.Provides good statistical properties, by avoiding the use of absolute values.

2.Use all the data

1.Not resistant to outliers.

2.Variance depends on the units of measurement, therefore not easy to make comparisons.

Sum of Squares

Measures variability of the scores, the total variation of all scores

1.Interval/ratio2.When no outliers

exist

1.Effect size calculation

1.Not resistant to outliers.

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Measures of Variability5

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Variance and Sum of Squares

2xxSS

1

2

2

n

xxS

1

2

n

xxS

x xx 2xx 6 1 1 5 0 0 3 -2 4 5 0 0 6 1 1

Mean = 5

Empirical Rule

The empirical rule states that symmetric or normal distribution with population mean μ and standard deviation σ have the following properties.

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Outcome Ball 1 Ball 2 Mean

1 1 1 1.0

2 1 2 1.5

3 1 3 2.0

4 2 1 1.5

5 2 2 2.0

6 2 3 2.5

7 3 1 2.0

8 3 2 2.5

9 3 3 3.0

All possible outcomes are shown below in Table 1.

Table 1. All possible outcomes when two balls are sampled with replacement.

Sampling Distribution

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Sampling Error

As has been stated before, inferential statistics involve using a representative sample to make judgments about a population. Lets say that we wanted to determine the nature of the relationship between county and achievement scores among Texas students. We could select a representative sample of say 10,000 students to conduct our study. If we find that there is a statistically significant relationship in the sample we could then generalize this to the entire population.

However, even the most representative sample is not going to be exactly the same as its population. Given this, there is always a chance that the things we find in a sample are anomalies and do not occur in the population that the sample represents. This error is referred as sampling error.

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Sampling Error

A formal definition of sampling error is as follows:Sampling error occurs when random chance produces a sample statistic that is not equal to the population parameter it represents.

Due to sampling error there is always a chance that we are making a mistake when rejecting or failing to reject our null hypothesis.

Remember that inferential procedures are used to determine which of the statistical hypotheses is true. This is done by rejecting or failing to reject the null hypothesis at the end of a procedure.

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Sampling Distribution and Standard Error (SE)

https://www.youtube.com/watch?v=hvIDuEmWt2k

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Hypothesis Testing

Null Hypothesis Statistical Significance Testing (NHSST)

Testing p-values using statistical significance tests

Effect Size

Measure magnitude of the effect (e.g., Cohen’s d)

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Null Hypothesis Statistical Significance Testing

Statistical significance testing answers the following question:

Assuming the sample data came from a population in which the null hypothesis is exactly true, what is the probability of obtaining the sample statistic one got for one’s sample data with the given sample size? (Thompson, 1994)

Alternatively:

Statistical significance testing is used to examine a statement about a relationship between two variables.

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Hypothetical Example

Is there a difference between the reading abilities of boys and girls?

Null Hypothesis (H0): There is not a difference between the reading abilities of boys and girls.

Alternative Hypothesis (H1): There is a difference between the reading abilities of boys and girls.

Alternative hypotheses may be non-directional (above) or directional (e.g., boys have a higher reading ability than girls).

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Testing the Hypothesis

Use a sampling distribution to calculate the probability of a statistical outcome.

pcalc = likelihood of the sample’s result

pcalc < pcritical: reject H0

pcalc ≥ pcritical: fail to reject H0

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Level of Significance (pcrit)

Alpha level (α) determines:

The probability at which you reject the null hypothesis

The probability of making a Type I error (typically .05 or .01)

True Outcome in Population

Reject H0 is true

H0 is false

Observed Outcome

Reject H0 Type I error (α) Correct Decision

Fail to reject H0

Correct Decision

Type II error (β)

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Example: Independent t-test

Research Question: Is there a difference between the reading abilities of boys and girls?

Hypotheses:

H0: There is not a difference between the reading abilities of boys and girls.

H1: There is a difference between the reading abilities of boys and girls.

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Dataset

Reading test scores (out of 100)

Boys Girls

88 88

82 90

70 95

92 81

80 93

71 86

73 79

80 93

85 89

86 87

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Significance Level

α = .05, two-tailed test

df = n1 + n2 – 2

= 10 + 10 – 2 = 18

Use t-table to determine tcrit

tcrit = ±2.101

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Decision Rules

If tcalc > tcrit, then pcalc < pcrit

Reject H0

If tcalc ≤ tcrit, then pcalc ≥ pcrit

Fail to reject H0

-2.101 2.101

p = .025 p = .025

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Computations

Boys Girls

Frequency (N) 10 10

Sum (Σ) 807 881

Mean () 80.70 88.10

Variance (S2) 55.34 26.54

Standard Deviation (S) 7.44 5.15

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Computations cont.

Pooled variance

Standard Error

= 40.944

= 2.862

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Computations cont.

Compute tcalc

Decision: Reject H0. Girls scored statistically significantly higher on the reading test than boys did.

= -2.586𝑡=𝑋 1−𝑋 2

𝑆𝐸 𝑋1− 𝑋2

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Confidence Intervals

Sample means provide a point estimate of our population means. Due to sampling error, our sample estimates may not perfectly represent our populations of interest. It would be useful to have an interval estimate of our population means so we know a plausible range of values that our population means may fall within.

95% confidence intervals do this.

Can help reinforce the results of the significance test.

CI95 = ± tcrit (SE)

= -7.4 ± 2.101(2.862) = [-13.412, -1.387]

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Statistical Significance vs. Importance of Effect

Does finding that p < .05 mean the finding is relevant to the real world?

Not necessarily…

https://www.youtube.com/watch?v=5OL1RqHrZQ8

Effect size provides a measure of the magnitude of an effect

Practical significance

Cohen’s d, η2, and R2 are all types of effect sizes

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Cohen’s d

Equation:

Guidelines: d = .2 = small

d = .5 = moderate

d = .8 = large

Not only is our effect statistically significant, but the effect size is large.

= -1.16