# Social Statistics and Data Analysis-Lecture 12-Sociology-Dr David Hall, Lecture notes for Social Statistics and Data Analysis. Nipissing University

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From Descriptive to Inferential Statistics.Inferential Statistics, Descriptive Statistics, Normal Curve, Sampling distribution, Central Limit Theorem, Mean, Mode, Median, Statistical Estimation, Confidence Intervals,Soci...
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Slide 1

From Descriptive to Inferential

Statistics….

Tables, Graphs, & Measures of Central

Tendency/ Variability are Descriptive

Statistics.

Inferential Statistics estimate population

characteristics, test hypotheses, & measure

relationships.

The Normal Curve (and standard scores)

statistics….and add to the interpretation of

the standard deviation.

Why is the normal curve so important in

statistics?

A large number of “real world” variables are

normally distributed in samples &

populations.

The sampling distribution of several

statistics are normally distributed

regardless of how the variables are

distributed in the population!!

The above theorem (Central Limit Theorem)

populations can be made from sample data.

Key aspects of the normal curve:

Normal curve is symmetrical or bell-shaped.

Average (mean) is the most frequently

occurring value (mode), & the value that

splits the distribution in half (median)…so

that 50% of the cases have values

greater than the mean, and 50% of the

cases have values lower than the mean

(Mean = Mode = Median).

Assuming a variable is normally distributed

we can say more about the standard

deviation.

If the mean IQ is 100 and the standard deviation

is 25, 34% of our 1000 cases (340 people) will

have IQs between 75 & 100; 34% (340 people)

will have IQs between 100 & 125; and 68% (680

people) will have IQs between 75 & 125….within

1 standard deviation of the mean.

If mean IQ is 100 & standard deviation is 25…13% of 1000

cases (130 people) have IQs between 125 - 150 (1 - 2

standard deviations above the mean). A total of 47% (13%

+ 34%) have IQs between 100 - 150 or up to 2 SDs above the

mean. Likewise, 13% (130 people) have IQs between 75 -

50 or between 1 and 2 SDs below the mean, & 47% (470

people) have IQs between 50 - 100 or up to 2 SDs below the

mean. Finally, 95% (2 X 47%) or 950 people have IQs within

2 SDs of the mean or between 50 & 150.

Only 2.1% or 21 people have IQs between 150 & 175 (2-3

standard deviations above mean), while another 2.1% or 21

people have IQs between 50 & 25 (2-3 standard deviations

below mean). Adding everything up (2.1 + 13.5 + 34.1 + 34.1 +

13.5 + 2.1) we can see that 99% of our sample of 1000 (999

people) have IQs between 25 & 175….or within 3 standard

deviations of the mean IQ of 100!

The percentages associated with

areas under the normal curve can

also be interpreted as probabilities!!!

z = standard score

Xi = any raw score or value

Xbar = sample mean

S = sample standard deviation

Why convert original values or scores on

a variable into standard scores?

1. Converting variable scores into

standard scores allows us to compare

original scores from different

distributions.

2. Converting variable scores into

standard scores allows us to express

variable scores in probabilities.

Statistical Estimation

“A poll found 31% of voters would support

the Liberals if an election were held today,

35% would vote for the Conservatives, 15%

for the NDP. The remainder of those polled

were undecided. The results are based

on a sample of 1000 respondents and

are accurate to within 5 percentage

points 99 times out of 100.”

And surveys are not just limited to political

polling. A survey of sex in the U.S. found

that:

“The average number of times per week that

a married couple have sex is 2.3 times per

week. The results are accurate to plus

or minus .2 sex acts 99 times out of

100.”

Survey research gets information from

small, representative samples to make

populations…..this is statistical estimation.

The normal curve connects up descriptive

and inferential statistics. Fundamental

inferential statistics involve statistical

estimation (e.g., estimating population

proportions, percentages & means….using

proportions, percentages & means obtained

from small, representative samples.