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Lecture 3

SOAN 3120 Lecture Notes - Lecture 3: Standard Deviation, Normal Distribution, Statistical Inference


Department
Sociology and Anthropology
Course Code
SOAN 3120
Professor
Andrew Hathaway
Lecture
3

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Wednesday, September 21
Density Curves & ‘Normal’ Distributions
Density Curves
- Height of bars on a histogram show the number or percentage of observations in each interval
oSmoothed out distribution
- Density curves instead show the proportion of data falling in each interval, so that the total area of bars
adds up to 1.0
- Ex. Imagine a large dataset for a quantitative, continuous variable such as height
oA histogram of data will get smoother and smoother as the number of bars grows, to the point
that it can be closely approximated by a curve
- The resulting density curve has the following properties:
oThe total area under the density curve is 1.0
oThe proportion of data lying between any two values of the variable is the area under the curve
between those values
Mean, Median, Standard Deviation
- The median of a density curve is the value of the variable that has half the area below it and half the area
above it
- If the density curve is made of solid material, then the mean (μ) is the point at which the curve balance
- In a symmetric distribution, the mean and median coincide. In a skewed distribution, both the mean and
the median are pulled in the direction of the skew
oThe mean more so than the median
- The standard deviation of a density curve (σ) – like the standard deviation (s) of a set of observations – is a
kind of average distance from the mean
Normal Distribution
- Play a central role in statistical inference
- All are symmetric, single-peaked, and bell shaped
- There is a different normal distributions for every pair of values of (μ) and (σ) – abbreviated by writing N
(μ, σ)
- All normal distributions follow the “68-95-99.7% rule”:
o68% (about 2/3) of the observations fall within one standard deviation of the mean
o95% of the observations fall within two standard deviations of the mean
o99.7% of the observations fall within three standard deviations of the mean
The Standard Normal Distribution
- All normal distributions are the same if we measure in units of size (σ) around the centre of the
distribution (μ)
oChanging to these units is called standardizing
- To find areas under normal distributions, it would be impossible to have tables for all possible values of
(μ) and (σ)
- By standardizing, we need only one table – for the standard normal distribution N (0 [mean], 1 [measuring
the distance from that])
- Standardization changes any normal distribution – with any mean and standard deviation N (μ, σ) – into a
normal distribution with a mean of 0 and a standard deviation of 1 – N (0, 1)
- We standardize a variable by subtracting its mean and dividing by its standard deviation
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