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Chapter 3

School

University of GuelphDepartment

Sociology and AnthropologyCourse Code

SOAN 3120Professor

Andrew HathawayChapter

3This

**preview**shows half of the first page. to view the full**3 pages of the document.**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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