## Search in the document preview

**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)
**

**link up descriptive & inferential
**

**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)
**

**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
**

**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. C**onverting 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
**

**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. **