AP Stats Unit 1B

summarizing data

parameter

fixed value about a population, typically unknown

• ex. the mean length of all the fish in all of Lake Lewisville

statistic

value calculated from a sample

• ex. the mean length of the sample collected of fish from Lake Lewisville

mean

arithmetic average

• use μ to represent a population mean

• use bar x to represent a sample mean

derivations from mean in a data set should all…

add up to zero

resistant meaning

statistical values that are not affected by extreme values

• median is resistant

• mean is not resistant

in a symmetrical distribution, mean and median are; which one to use

equal; you should use the mean in sym dist.

in a skewed distribution, mean and median are; which one to use

the mean is pulled in the direction of the skewness; you should use the median in skew dist.

trimmed mean; how to calculate

purpose is to remove outliers from a data set

to calc a trimmed mean:

• multiply the percent to trim by n

• truncate that many observations from BOTH ends of the distribution (when listed in order)

• calculate the mean with the shortened with the shortened data set

value used to describe categorical data

p-hat (p̂) = x / n

used to represent the sample proportion; It measures the fraction or percentage of items in a specific sample that share a particular trait (called a "success")

the study of variability is important bc…

there is variability in virtually everything, allows us to distinguish between usual & unusual values, reporting only a measure of center doesn’t provide a complete picture of the distribution

simplest numeric measurement of variability

range

another measure of variability (using mean)

deviations from mean (x-x̄)

variance

estimated average of the deviations squared

• s² for sample variance (remember to n-1 in denominator)

• σ² for population variance

square root of variance = standard deviation

ex. s² = 6.28 in² ; s = 2.608 in

fish in sample can deviate from mean of 6 by an average of 2.608 inches

most common used measures of center and variability

mean and standard deviation, respectively

degrees of freedom (df)

the number of independent observations that are free to vary

ex. we consider the sample of 6 fish where the mean is 6 inches, five of these values are freed to be any possible length of fish; but once these five values occur, the sixth value is no longer free to vary (it must be a specific value in order for the derivations from the mean [of 6] to have a sum of zero)

sample of n; how to find DOF

n-1; n-1 observations are free to vary

IQR

the range of the middle half of the data: Q3 - Q1

IQR is akin to median as SD is to the mean

advantage of IQR to SD: IQR is resistant to extreme values

boxplot advantages

• ease of construction, convenient handling of outliers, construction is not subjective (like histograms)

• used with medium or large size data sets (n ≥ 10)

• useful for comparative displays

boxplots overview

when to use: univariate numerical data; moderate to large data sets (no under 10s)

how to construct (skeleton boxplot)

• calculate the five number summary

• draw a horizontal (or vertical scale)

• construct a rectangular box from the lower quartile (Q1) to the upper quartile (Q3)

• divide the box with a line representing the median

• draw lines from the lower quartile to the smallest observation and from the upper quartile to the largest observation

to describe

• comment on the center, spread, and shape of the distribution and if there are any unusual features

identifying and displaying outliers (modified boxplots)

1.5(IQR) method: outlier exists if it is more than 1.5(IQR) away from the nearest quartile

• Q1 - 1.5(IQR) and Q3 + 1.5(IQR)

• resulting values are the “fences” and any value above the upper fence or below the lower fence are outlies

whiskers extend to largest (or smallest) data observation that is not an outlier

• mark outlier as asterisk

symmetrical boxplots vs approx symmetrical boxplot vs skewed

sym: you can’t tell how many modes there are

aprox sym: if the lower half of the range and the upper half of the range in the distribution is approx equal, it is approx symmetrical

skewed: range of the two halves definitely different sizes, so it would be skewed in the direction of the longest side

example CUSS and BS for comparing distributions

The Laker’s salaries have a higher median ($5M) than the Nuggts ($3M), and a larger range than the Nuggets (~$20M is greater than $15M). Both salary distributinos are skewed right, the Lakers have an outlier at ~23 while Nuggets have no outliers

• remember comparative language and values

example FRQ talking about outliers

Yes, 47% of residents having a college degree is an outlier since it is larger than the upper fence therefore is an outlier, there are no values below the lower fence

Normal Curve

• bell shaped, symmetrical, unimodal curve

• transition points happen cupping upward and downward occur at μ ± σ

• as the standard deviation increases, the curve flattens and spreads

• as the standard deviation decreases, the curve gets taller and thinner

empirical rule (only for normal distributions [bell])

approx 68% of the observations are within 1 SD of the mean

approx 95% of the observations are within 2 SD of the mean

approx 99.7% of the observations are within 3 SD of the mean

z score

z-score = (value-mean)/SD

z score tells us how many standard deviations the value is from the mean

• ex. -2.3 is 2.3 standard deviations below the mean

• 1.8 is 1.8 SD above mean

percentiles - measures of relative standing

a percentile is a value in the data set where r percent of the observations fall AT or BELOW that value

linear transformation rule

• when adding a constant to a random variable, the mean changes but not the standard deviation

• when multiplying a constant to a random variable, the mean and the SD changes

rules for combining two variables

• to find the mean for the sum (or difference), add (or subtract) the two means

• to find the standard deviation of the sum (or differences), always add the variances (squared version), then take the square root