We now have a toolbox of graphical and numerical tools for describing distributions of data, along with a strategy:

1.Start with a graph, usually a stemplot or histogram

2.Look for the overall patterns and striking deviations like outliers

3.Choose a numerical summary to briefly describe center and spread.

Add 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

Example: Toss a coin 20 times, record the proportion of heads:

Mark your result on the board

Def: Distribution - What values can occur and how often each value does occur/the pattern (p. 5).

Def: Frequency Curve - Smoothed out version of the overall pattern, not irregularities or outliers, (approximation or idealization, a mathematical model).

Coin tossing: Idealized mean=m=.5, Idealized standard deviation=s=.11

Def: Relative Frequency/Density curve - Height = proportion of times the value occurs. Relative Frequency = number of times occurred/total observations. Heights all add up to 1.

Shape is identical to shape of frequency curve.

Most Important Density Curve=Normal Curve N(m, s)

Features: Described by:

Note: s denotes where curvature changes.

Gives an overall description of the data but must check this assumption! Not everything is Normal!

In any Normal Distribution with mean m and standard deviation s

• of the observations lie within standard deviation of m
• of the observations lie within standard deviations of m
• of the observations lie within standard deviations of m

Prediction: Almost all of you would get between .28 and .72

More often than not between .39 and .61!

Example 1 Do the proportions follow the 68-95-99.7 Rule?

Interval % in interval % if normal

(-s, +s)=

(-2s, +2s)=

(-3s,+3s)=

Example 2 The distribution of heights of adult men is considered to be Normal with mean m=69 inches and standard deviation s=2.5 inches.

- Sketch this Normal curve

- Between what heights do the middle 68%/ middle 95% of men fall?

- What percent of men are taller than 74 inches?

- What percent of men are shorter than 66.5 inches?

- What percent of men are shorter than 65 inches?

Example 3 How many standard deviations is a height of 65 inches from the mean?

• Indicates how far (distance!) observation lies from the mean and direction
• Positive if observation lies above the mean, negative if lies below.
• Converts all observations to a common scale

All Normal distributions can be standardized!

Example 4 Eleanor scores 680 on the Math SAT and Gerald scores a 27 on Math ACT, who has the higher score?

SAT:N(500,100) ACT:N(18,6)

Percentiles: What percent of the population falls below this value?

Table A contains the percentiles for Standard Scores

To find the percentile of an observed value:

1. Standardize the score (subtract mean, divide by std dev)

2. Look up the percentile in Table A

Example 5 What percent of men are shorter than 65 inches?