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Chapter 3

SOAN 3120 Chapter Notes - Chapter 3: Abbreviation, Standard Scale, Standard Deviation


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

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Chapter 3 – The Normal Distributions
3.1 – Density Curves
- A density curve is a curve that:
oIs always on or above the horizontal axis
oHas area exactly 1 underneath it
- A density curve describes the overall pattern of a distribution
3.2 – Describing Density Curves
- Measures of center and variability apply to density curves
- The median is the point with half the observations on either side
oThe median of a density curve is the equal-areas point, the point with half the area under the
curve to its left, and the remaining half of the area to its right
- The quartiles divide the area under the curve into quarters
- A symmetric density curve is exactly symmetric
oThe median of a symmetric density curve is therefore at its center
- It isn’t always easy to spot the equal-areas point on a skewed curve
- If we think of the observations as weights strung out along a thin rod, the mean is the point at which the
rod would balance
oThe mean is the point at which the curve would balance if made of solid material
- The mean and median of a symmetric density curve are equal
- On a skewed density curve, the mean is pulled toward the long tail more than the median
- Because a density curve is an idealized description of a distribution of data, we need to distinguish
between the mean and the standard deviation of the density curve, and the mean x̄ and standard
deviation s computed by the actual observations
oThe mean of a density curve is µ
oThe standard deviation of a density curve is σ
3.3 – Normal Distributions
- Normal curves describe normal distributions
oDoesn’t necessarily mean usual or average
- Several important facts:
oAll normal curves have the same overall shape: symmetric, single-peaked, bell-shaped
oAny specific normal curve is completely described by giving its mean µ and its standard deviation
σ
oThe mean is located at the center of the symmetric curve and is the same as the median
Changing µ without changing σ moves the normal curve along the horizontal axis
without changing its variability
oThe standard deviation σ controls the variability of a normal curve
When the standard deviation is larger, the area under the normal curve is less
concentrated about the mean
- One can completely determine the shape of a normal curve using µ and σ, and can also located σ by eye
on a normal curve
oThe point at which the change of curvature takes place are located at distance σ on either side of
the mean µ
This only works for normal distributions
- Why are normal distributions important?
1. Normal distributions are good descriptions for some distributions of real data, such as scores on tests
2. Normal distributions are good approximations to the results of many kinds of chance outcomes, such
as the proportion of heads in many tosses of a coin
3. We will see that many statistical inference procedures based on normal distributions work well for
other roughly symmetric distributions
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