Chapter 3

1. Section 3.1: Measures of Central Tendency (also see Palay: Central Tendency 17:56 (script).)
   • Mean
   • Notation
     • List of `n` values is given as `x_1, x_2, x_3, ... ,x_(n-1), x_n`.
     • We use the capital Greek Letter sigma, `Sigma`, to mean "the sum of"
     • The complete notation might be given as `Sigma_(i=1)^n x_i` meaning the sum of the things we call `x_i` where `i` takes on the values from `1` to `n`. We could wxpress this as `Sigma_(i=1)^n x_i = x_1 + x_2 + x_3 + ... + x_(n-1) + x_n`
     • The sloppy, but still understandable, notation is given as `Sigma x`
     • We use the lower case Greek letter mu, `mu`, to represent the mean of a population.
     • A population with `N` values has a mean defined as `mu = (x_1 + x_2 +x_3 + ... + x_(N-1) + x_N)/N = (Sigma_(i=1)^N x_i)/N = (Sigma x)/N`
     • We use the symbol `x` with a bar over it, `bar x`, to represent the mean of a sample.
     • A sample with `n` values has a mean defined as `bar x = (x_1 + x_2 +x_3 + ... + x_(n-1) + x_n)/n = (Sigma_(i=1)^n x_i)/n = (Sigma x)/n`
   • Median
     • Funny rule if there are an even number of values, namely, "use the average of the two middle values"
     • Why the "funny rule" is not important in the real world.
     • Why the "funny rule" is very important in math tests.
   • Influence of extreme values.
   • A statistic is resistant if its value is not affected "much" by extreme values in the data set.
   • Skew and symmetry
     • median < mean is called "skewed to the right"
     • mean < median is called "skewed to the left"
     • mean and median are "close" to the same value, then we cannot call the shape skewed in either direction, so we say it is symmetric.
     • Hint: skewed is in the direction of the long tail.
   • Mode
     1. Expanded stem-leaf worksheet with median and mode
     2. Expanded stem-leaf worksheet with mean, median, and mode
   • Approximating the Mean with Grouped Data (you may want to look at the calculator support page given as a link in subtopic 4 of this page).
2. Section 3.2: Measures of Spread (Dispersion)
   • Range
   • Variance (population and sample)
     • We use the square of the lower case greek letter sigma, `sigma^2`, to represent the variance of a population.
     • The defining equation for the population variance is: `sigma^2 = (Sigma_(i=1)^N (x_i - mu )^2)/N = (Sigma(x - mu )^2)/N`
     • An equivalent and easier to compute equation is `sigma^2 = (Sigma_(i=1)^N ((x_i)^2) - N mu^2)/N = (Sigma x^2 -N mu^2)/N`
   • Standard Deviation (population and sample) (also see Palay: The Standard Deviation 15:12 (script).)
   • Standard Deviation not resistant
   • Approximating Variance (Standard Deviation) with Grouped Data
   • Bell shape
   • Empirical Rule
     • About 68% of all values within 1 standard deviation of meaan.
     • About 95% of all values within 2 standard deviations of mean.
     • About 99% of all values within 3 standard deviations of mean.
     • Evenly balanced, so 34% up and 34% down 1 standard deviation, etc.
   • Chebyshev's Inequality: at least `(1-1/K^2)` of the values in a data set will be within `K` standard deviations of the mean.

     Number of standard deviations from the mean K	Chebyshev's Proportion	Empirical Proportion for bell shaped distributions
     1	`1-1/1^2 = 1-1 = 0`	`68%`
     2	`1-1/2^2 = 1-1/4 = 75%`	`95%`
     3	`1-1/3^2 = 1-1/9 = 88.9%`	`99%`
     4	`1-1/4^2 = 1-1/16 = 93.75%`	`99.999%`

   • Coefficient of variation `CV = sigma/mu` or `CV =(\quad s \quad )/(bar x)`
3. Section 3.3
   • `z`-score of a value in a population: `z = (x - mu)/sigma`
   • `z`-score and the Empirical Rule (for what look like normal, bell shaped, distributions)
   • Percentiles: (also see Palay: Percentiles 16:36 (script).)
     • From percentile find data item.
       for `n` items in ascending order, `L= p/100*n`; if `L` whole then average item(`L`) and item(`L+1`), else round-up to `L^+` and use item(`L^+`)
     • From data item find percentile.
       for `n` items, put them in ascending order, find `m`=number of items less than the data value, compute `P=100((m+0.5)/n)` and round up
   • Quartiles
     1. Palay: Quartile Points 17:36 (script).
     2. Worksheet with mean, median, mode, and quartiles
   • five-number summary
     

     Minimum	First quartile	median	Third quartile	Maximum
     Q0	Q1	Q2	Q3	Q4
     0%	25%	50%	75%	99%

   • Outliers
   • Interquartile Range: IQR =Q₃ – Q₁
     Lower outlier boundary = Q₁ – 1.5*IQR
     Upper outlier boundary = Q₃ + 1.5*IQR
   • Box and Whisker Plots
     1. Worksheet on Box and Whisker Plots: Part I
     2. Worksheet on Box and Whisker Plots: Part II
4. Calculator support
   • General Use: 1-Var Stats, Box-and-whisker, Histograms, COLLATE3.
   • Discrete General Use: 1-Var Stats, Histogram, COLLATE2, Group Data, Box-and-whisker.
   • Group Data: Group Data, 1-Var Stats, GRPAPPX, Histogram, COLLATE3.