Chapter 2 – Descriptive Statistics

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Stemplot

Histogram (bar graph)

Frequency Polygon (line graph)

Measures of Location

• Quartiles divide datasets into four equal quarters

• Percentiles divide datasets into hundredths

  • $i=\frac{k}{100}(n+1)$ where $i$=index, $k$=$k^{th}$ percentile, $n$=total datapoints
  • $\text{percentile}=\frac{x+0.5y}{n}(100)$ where $x$=datapoints from bottom to datapoint (exclusive), $y$=datapoints equal to measured datapoint, $n$=total datapoints

• Median refers to the middle datum in an ordered dataset (simple average of two middle points if even number of data)

• Interquartile range (IQR) refers to the spread between the first and third quartiles

  • $\text{IQR}=Q_3-Q_1$
  • Outliers are $\pm 1.5\text{ IQR}$ from the 50^th percentile (median)

Box Plots

Measures of Center

• Mean refers to the simple average

  • $\text{mean}=\frac{\Sigma x_i}{n}$
  • $\bar{x}$ is the sample mean
  • $\mu$ is the population mean

• Median refers to the middle value

  • $\text{index}=\frac{n+1}{2}$

• Mode refers to the most frequent value (a dataset with >1 mode is bimodal)

Law of Large Numbers

$\bar{x}\to \mu$ as sample size increases.

Sampling Distributions

• Sampling distributions show probability of every result for a statistic from a large sample
  • Statistics are calculated from samples ($\bar{x}$ is a statistic from a sample that estimates $\mu$)

Grouped Frequency Tables

• The mean of a grouped frequency table can be calculated as $\frac{\Sigma (f\cdot m)}{\Sigma f}$ where $f$=interval frequency and $m$=interval midpoint

Skewness

• Symmetrical distributions have the same mean and median
• Left-skewed distributions have a longer left tail (mean < median)
• Right-skewed distributions have a longer right tail (mean > median)

Measures of Spread

Standard Deviation

• Standard deviation measures how far values are spread from the mean

  • $s$ is the sample standard deviation
  • $\sigma$ is the population standard deviation

• Deviation refers to the difference between some number $x$ and the mean ($x-\bar{x}$ or $x=\sigma$)

• Variance refers to the average of the square of the deviations

  • Sample variance: $s^2=\frac{\Sigma (x-\bar{x}{)}^2}{n-1}$
  • Population variance: ${\sigma }^2=\frac{\Sigma (x-\mu {)}^2}{n}$
  • Therefore, standard deviation is the square root of variance

Sampling Variability

• Sampling variability refers to how much a statistic varies between samples
  • Standard error of the mean is a standard deviation that measures sampling variability ($\frac{\sigma }{\sqrt{n}}$)

Comparing Values

• The $z$-score is a metric that compares values from different datasets by standard deviations
  • Sample: $x=\bar{x}+zs$, $z=\frac{x-\bar{x}}{s}$
  • Population: $x=\mu +z\sigma$, $z=\frac{x-\mu }{\sigma }$

Chebyshev’s Rule

• For any dataset regardless of distribution
  • $\ge$ 75% are within 2 SD
  • $\ge$ 89% are within 3 SD
  • $\ge$ 95% are within 4.5 SD

Empirical Rule

• For symmetric, bell-shaped distributions
  • $\approx$ 68% are within 1 SD
  • $\approx$ 95% are within 2 SD
  • $\approx$ 99% are within 3 SD