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Statistics 270 - Lecture 3

- Last class types of quantitative variable,

histograms, measures of center, percentiles and

measures of spreadwell, we shall finish these

today - Will have completed Chapter 1
- Assignment 1 Chapter 1, questions 6, 20b, 26,

36b-d, 48, 60 - Some suggested problems
- Chapter 1 1, 5, 13 or 14 (DO histogram), 19, 26,

29, 33

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Measures of Spread (cont.)

- 5 number summary often reported
- Min, Q1, Q2 (Median), Q3, and Max
- Summarizes both center and spread
- What proportion of data lie between Q1 and Q3?

Box-Plot

- Displays 5-number summary graphically
- Box drawn spanning quartiles
- Line drawn in box for median
- Lines extend from box to max. and min values.
- Some programs draw whiskers only to 1.5IQR above

and below the quartiles

- Can compare distributions using side-by-side

box-plots - What can you see from the plot?

Other Common Measure of Spread Sample Variance

- Sample variance of n observations
- Can be viewed as roughly the average squared

deviation of observations from the sample mean - Units are in squared units of data

Sample Standard Deviation

- Sample standard deviation of n observations
- Can be viewed as roughly the average deviation of

observations from the sample mean - Has same units as data

Exercise

- Compute the sample standard deviation and

variance for the Muzzle Velocity Example

- Variance and standard deviation are most useful

when measure of center is - As observations become more spread out, s

increases or decreases? - Both measures sensitive to outliers
- 5 number summary is better than the mean and

standard deviation for describing (I) skewed

distributions (ii) distributions with outliers

Population and Samples

- Important to distinguish between the population

and a sample from the population - A sample consisting of the entire population is

called a - What is the difference between the population

mean and the sample mean? - The population variance ( or std. deviation) and

that of the population - Population median and sample median?

Empirical Rule for Bell-Shaped Distributions

- Approximately
- 68 of the data lie in the interval
- 95 of the data lie in the interval
- 95 of the data lie in the interval
- Can use these to help determine range of typical

values or to identify potential outliers

ExamplePutting this all together

- A geyser is a hot spring that becomes unstable

and erupts hot gases into the air. Perhaps the

most famous of these is Wyoming's Old Faithful

Geyser. - Visitors to Yellowstone park most often visit Old

Faithful to see it erupt. Consequently, it is of

great interest to be able to predict the interval

time of the next eruption.

ExamplePutting this all together

- Consider a sample of 222 interval times between

eruptions (Weisberg, 1985). The first few lines

of the available data are - Goal Help predict the interval between

eruptionsConsider a variety of plots that may

shed some light upon the nature of the intervals

between eruptions

ExamplePutting this all together

- Goal Help predict the interval between eruptions
- Consider a histogram to shed some light upon the

nature of the intervals between eruptions

ExamplePutting this all together

ExamplePutting this all together

- What does the box-plot show?
- Is a box-plot useful at showing the main features

of these data? - What does the empirical rule tell us about 95 of

the data? Is this useful? - We will come back to this in a minute

Scatter-Plots

- Help assess whether there is a relationship

between 2 continuous variables, - Data are paired
- (x1, y1), (x2, y2), ... (xn, yn)
- Plot X versus Y
- If there is no natural pairingprobably not a

good idea! - What sort of relationships might we see?

ExamplePutting this all together

- What does this plot reveal?

ExamplePutting this all together

ExamplePutting this all together

- Suppose an eruption of 2.5 minutes had just taken

place. What would you estimate the length of the

next interval to be? - Suppose an eruption of 3.5 minutes had just taken

place. What would you estimate the length of the

next interval to be?

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