AP Stat Chapter 5 Vocab: The Standard Deviation as a Ruler and the Normal Model

Normal model

A useful family of models for unimodal, roughly symmetric distributions. (p. 114) *Bell-shaped curves are called this

Statistic

A value calculated from data to summarize aspects of the data. For example, the mean, ȳ , and standard deviation, s, are statistics. (p. 114)

Standardized value

A value found by subtracting the mean and dividing by the standard deviation. (p. 108)

z-score

A z-score tells how many standard deviations a value is from the mean, and in which direction; z-scores have a mean of 0 and a standard deviation of 1. When working with data, use the statistics ȳ and s: z = (y-ȳ)/s When working with models, use the parameters µ and σ: z = (y-µ)/σ (p. 114)

Shifting

Adding (or subtracting) a constant to each data value adds (or subtracts) the same constant to the measures of position (such as mean, median, quartiles, percentiles, min, max), but does not change the measures of spread (range, standard deviation, and IQR). (p. 111)

Assessing Normality

1. Determine if 68-95-99.7 rule applies 2. Look at the histogram. 3. Look at the Normal Probability Plot. 4. Compare the mean and median. Figure out what number is +3 SD above mean and see if it is close to the max and do the same for min.

Where to find normal distribution on calculator

2nd Distr; most often use normalcdf(zLeft, zRight, 0, 1) and invNorm(area, 0, 1) *See pg. 119-120

Parameter

A numerically valued attribute of a model. For example, the values of µ and σ in a N(µ,σ) model are parameters. (p. 114) *This mean and standard deviation are not numerical summaries of the data (they are part of the model, don't come from the data, we chose them)

How to sketch a normal curve

To sketch a good Normal curve, you need to remember only three things: ■ The Normal curve is bell-shaped and symmetric around its mean. Start at the middle, and sketch to the right and left from there. ■ Even though the Normal model extends forever on either side, you need to draw it only for 3 standard deviations. After that, there's so little left that it isn't worth sketching. ■ The place where the bell shape changes from curving downward to curving back up—the infection point—is exactly one standard deviation away from the mean.

Normality Assumption

We must have a reason to believe a variable's distribution is Normal before applying a Normal model. (p. 115)

Rescale

Multiplying (or dividing) each data value by a constant multiplies (or divides) both the measures of position (such as mean, median, quartiles, percentiles, min, max) and the measures of spread (range, standard deviation, and IQR) by that constant. (p. 111)

Standard Normal model

A Normal model, N(µ,σ) with mean µ = 0 and standard deviation σ = 1. Also called the standard Normal distribution. (p. 114) Standardizing into z-scores: - Does not change the shape - Changes the center by making the mean 0 - Changes the spread by making the standard deviation 1 *For symmetric data, the standard deviation is usually a bit smaller than the IQR, and it's not uncommon for at least have of the data to have z-scores between -1 and 1. *No matter the shape of a distribution, a z-score of 3 (plus or minus) or more is rare and a z-score of 6 or 7 shouts out for attention. (see pg. 118 for rule) *Any observation +/- 2 SDs from the mean is considered unusual for normal model *Any observation +/- 3 SD from the mean is considered extremely unsual/rare for a normal model

Normal probability plot

A display to help assess whether a distribution of data is approximately Normal. If the plot is nearly straight, the data satisfy the Nearly Normal Condition. (p. 125)

Nearly Normal Condition

A distribution is nearly Normal if it is unimodal and symmetric. We can check by looking at a histogram (or a Normal probability plot). (p. 115)

68-95-99.7 Rule

In a Normal model, about 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations of the mean, and about 99.7% fall within 3 standard deviations of the mean. (p. 115)

Normal percentile

The Normal percentile corresponding to a z-score gives the percentage of values in a standard Normal distribution found at that z-score or below. (p. 119)

cumulative frequency graph

The total of a frequency and all frequencies so far in a frequency distribution. It is the 'running total' of frequencies.