From Descriptive to Inferential
• Tables, Graphs, & Measures of Central
Tendency/ Variability are Descriptive
• Inferential Statistics estimate population
characteristics, test hypotheses, & measure
• The Normal Curve (and standard scores)
link up descriptive & inferential
statistics….and add to the interpretation of
the standard deviation.
Why is the normal curve so important in
• A large number of “real world” variables are
normally distributed in samples &
• 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)
is why accurate inferences about
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
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
2. Converting variable scores into
standard scores allows us to express
variable scores in probabilities.
• “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
“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
• Survey research gets information from
small, representative samples to make
accurate generalizations about large
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.
From Estimation to Confidence Intervals….
• In repeated sampling, the distribution of
means is normally distributed with a mean
equal to the true population mean.
• We can define a range of sample mean
values within which the true population
mean is likely to be & estimate probability
our range of sample means includes the
population mean…this is the concept of
the confidence interval.