Chapter 11

Central TendencyDispersion

Statistical InferenceHypothesis Testing

Description• We can describe data in a number of ways:–We could describe every observation, or every

value in a data set (but this would be overwhelming and mostly unhelpful)

– Alternatively, we could summarize the data:• Graphical summaries

– Bar graphs, pie graphs, dot plots, etc.

• Statistical summaries– Frequency distributions– Descriptive statistics

Description• Frequency distributions– A table that shows the number of observations

having each value of a variable–May include other statistics like the relative

frequency proportion, percentage, missing values, or odds ratios

• Descriptive statistics– Describing a large amount of data with just one

number

Description

• Two classes of descriptive statistics– Central tendency– Dispersion

Central Tendency

• Measures of central tendency– Describe the typical case in a data set or

distribution– Three statistics• Mode• Median• Mean

Central Tendency

• Mode– Indicates the most common observation– Simply count the number of times you observe

each value–Mode is resistant to outliers• By definition, the mode cannot be an outlier• Describes only a single value in the data

Central Tendency• Median– Describes the middle value in an ordered set of

values– Important to rank order the observations first–Median = (N+1)/2 –With an even number of observations, average

the two middle values– Resistant to outliers—by definition, median is not

an outlier– Includes only one value

Central Tendency

• Mean– Describes the average value–Mean = (∑Y)/N–Mean is not resistant to outliers• Outliers will pull the mean up or down, sometimes

significantly

– Computed using all values

Central Tendency• Compute the mode, median, and mean for each

of these data sets:Data set #1 Data set #2

i Y i Y1 5 1 12 5 2 43 5 3 54 5 4 55 5 5 10

Central Tendency

• Data set #1– Mode = 5– Median = 5– Mean = 5

• Clearly, the two data sets are not identical

• Data set #2– Mode = 5– Median = 5– Mean = 5

• But central tendency belies the truth

Dispersion

• What we need is some way to differentiate between data set #1 and data set #2.

• The typical values in each data set were the same.

• We need a measure that describes the other values in the data sets.

• Measures of dispersion indicate how the other values vary around the typical value.

Dispersion

• Measures of dispersion– Range– Variance– Standard deviation

Dispersion

• Range– One of the simplest measures of dispersion is the

range.– Range = Y maximum – Y minimum – Describes the extremes of the data around the

typical case.

Dispersion

• Variance– The variance takes into account all of the values in the

data set.– There are two formulas to calculate the variance:

• One formula for the sample• One formula for the population

– The only difference is that we subtract 1 from the sample size in the sample version of the equation.

Dispersion• Standard deviation– The standard deviation also takes into account all of

the values in the data set.– There are also two formulas to calculate the standard

deviation:• One formula for the sample• One formula for the population

– Like variance, the only difference is that we subtract 1 from the sample size in the sample version of the equation.

Dispersion• Compute the range, sample variance, and sample

standard deviation for each of these data sets:Data set #1 Data set #2i Y i Y1 5 1 12 5 2 43 5 3 54 5 4 55 5 5 10

Dispersion

• Data set #1– Range = 0– Variance = 0– Standard deviation = 0

• Measures of dispersion indicate that the data sets are not the same.

• Data set #2– Range = 9– Variance = 10.5– Standard deviation = 3.24

Dispersion

• Now try calculating the population versions of the variance and standard deviation for data set #2.

• Data set #2– Variance = ?– Standard deviation = ?

Dispersion

• As you can see, the population variance and standard deviation are slightly smaller than in the sample version.

• This reflects our greater confidence in population data than in sample data.

• Data set #2– Variance = 8.4– Standard deviation = 2.89

Dispersion• Variance and standard deviation– Variance is used in many different statistical

applications.– The standard deviation is used more often to

summarize the data than variance because the standard deviation is in the same units as the mean.

– If data sets #1 and #2 describe miles per gallon, we could say that in data set #2 we have a mean of 5 miles per gallon and a standard deviation of 2 miles per gallon.

Statistical Inference– The normal distribution is our first choice in most

cases because it has such wonderful properties:• Distribution is symmetrical around the mean• Percentage of cases associated with standard deviations• Can identify probability of values under the curve• A linear combination of normally distributed variables is

itself distributed normally• Central limit theorem • Normal distribution is symmetric and mesokurtic

– Great flexibility in using the normal distribution

Statistical Inference

Statistical Inference

• We can calculate a z score for every observation in the data set.

• The z score allows us to compare each observation to the rest of the data set, relative to the mean.

• z score, or z of X = (X – )

Statistical Inference

• Example:• = 64 = 2.4 Xi=70 or more

• z = (X – ) /

Statistical Inference

• Example:• = 64 = 2.4 Xi=70 or more

• z = (X – ) / • z = (70 – 64) / 2.4• z = (6) / 2.4• z = 2.5 for 70 contacts• p = .0062; or 0.62%

Hypothesis Testing

• How do you test hypotheses with statistics?• Comparing the means of two groups– Consider an experiment

• Research hypothesis: • Null hypothesis:

X1 ≠ X2 ──

X1 = X2

─ ─

Hypothesis Testing

• Type 1 error– State of the world: Research hypothesis is false– Incorrect rejection of null

• Type 2 error– State of the world: Research hypothesis is true– Incorrect acceptance of null

Hypothesis Testing

• Hypothesis: College students are less likely to read political news stories than are other voting-age citizens.

• X = 5; = 10; = 2; n = 25

(X – )

( / √n)__________z =

_

_

Hypothesis Testing

• Hypothesis: College students are less likely to read political news stories than are other voting-age citizens.

• X = 5; = 10; = 2; n = 25

(X – )

( / √n)__________z =

_

_

(5 – 10)

(2 / √25)__________z =

(-5)

(.4)__________z =

-12.5 z =

Hypothesis Testing

• Hypothesis: College students are less likely to read political news stories than are other voting-age citizens.

• 95% confidence• z critical = 1.96

(X – )

( / √n)__________z =

_

(5 – 10)

(2 / √25)__________z =

(-5)

(.4)__________z =

-12.5 z =

Hypothesis Testing

• Hypothesis: College students rate liberal candidates higher than do the rest of the voting population.

• X = 52; = 50; = 5; n = 25

(X – )

( / √n)__________t =

_

_

Hypothesis Testing

• Hypothesis: College students rate liberal candidates higher than do the rest of the voting population.

• X = 52; = 50; = 5; n = 25

(X – )

( / √n)__________t =

_

(52 – 50)

(5 / √25)___________t =

(2)

(1)__________t =

2 t =

_

Hypothesis Testing

• Hypothesis: College students rate liberal candidates higher than do the rest of the voting population.

• Two-tailed test; .05 level; n – 1 df

• t critical = 2.064

(X – )

( / √n)__________t =

_

(52 – 50)

(5 / √25)___________t =

(2)

(1)__________t =

2 t =