In doomlab/learnSTATS: Introduction to Statistics (ANLY 500)

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
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Breaking Down the Research Process: A Review

• To analyze the data and generate interpretable results the following statistical models can be used:

  • Frequency distribution, measures of shape
  • Central tendencies, measures of distance
  • Dispersion, measures of spread
  • Going beyond the data with z-scores and association.

Frequency Distributions

• Definition - A graph plotting values of observations on the horizontal axis, with a bar showing how many times each value occurred in the data set.
• These are measures of shape or histograms

Frequency Distributions

• Understanding the distribution of the data can be done by looking at its shape, for instance:

• The 'Normal' Distribution

  • Bell-shaped
  • Symmetrical around the center

• Normal Distribution Visualized

knitr::include_graphics("pictures/introDA2/ndis.jpg")

Frequency Distributions

• Frequency tables are useful to understand the distribution of data.
• In R, you can capture a frequency table of the quakes dataset:

library(datasets)
data("quakes")
str(quakes)

Frequency Distributions: Tables

• You will learn that the table() function comes in quite handy to create a frequency table.

table(quakes$mag)

Frequency Distributions: Plots

• Create a plot of the distribution:

hist(quakes$mag, breaks = 24)

Understanding Frequency Distributions

• Frequency distribution properties include the following statistics:

  • Skew - The symmetry of the distribution.

    • Positive skew (scores bunched at low values with the tail pointing to high values).
    • Negative skew (scores bunched at high values with the tail pointing to low values).

  • Kurtosis - The 'heaviness' of the tails.

    • Leptokurtic = heavy tails.
    • Platykurtic = light tails.

Skew Visualized

knitr::include_graphics("pictures/introDA2/Skew_PosvsNeg.png")

Kurtosis Visualized

knitr::include_graphics("pictures/introDA2/Kurtosis.png")

Produce a Histogram

• Now that we know these terms...what shape does our histogram have?

hist(quakes$mag)

Calculating Skewness

library(moments)
skewness(quakes$mag)

library(psych)
describe(quakes$mag)

Interpreting Skewness

• How can you interpret the skewness number?

• Bulmer (1979) - a classic - suggests this rule of thumb:

  • If skewness is less than -1 or greater than +1, the distribution is highly skewed.
  • If skewness is between -1 and -0.5 or between +0.5 and +1, the distribution is moderately skewed.
  • If skewness is between -0.5 and +0.5, the distribution is approximately symmetric.

Calculating Kurtosis

#moments
kurtosis(quakes$mag)

#psych
describe(quakes$mag)

Interpreting Kurtosis

• How can you interpret the kurtosis number?

  • The reference standard is a normal distribution, which has a kurtosis of 3.
  • However, its common to consider the excess kurtosis, which is simply kurtosis -3.

Interpreting Excess Kurtosis

• Excess Kurtosis can be interpreted using the following rule of thumb:

  • A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0, 3-3).
  • Any distribution with kurtosis 3 (excess 0) is called mesokurtic.
  • A distribution with kurtosis < 3 (excess kurtosis < 0) is called platykurtic.
  • Compared to a normal distribution, its tails are shorter and thinner, and often its central peak is lower and broader.
  • A distribution with kurtosis > 3 (excess kurtosis > 0) is called leptokurtic.
  • Compared to a normal distribution, its tails are longer and fatter, and often its central peak is higher and sharper.

• We will cover more on skew and kurtosis in the data screening section.

Central Tendency

• Definition - A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data.
• As such, measures of central tendency are sometimes called measures of central location.
• They are also called summary statistics.

summary(quakes$mag)

Other Summary Stat Functions

• The pastecs package.

library(pastecs)
stat.desc(quakes)

Other Summary Stat Functions

• The Hmisc package.

library(Hmisc)
Hmisc::describe(quakes)

Other Summary Stat Functions

• The psych package.

library(psych)
psych::describe(quakes)

The Mean

• Definition - The sum of scores divided by the number of scores.

$$\bar{x} = \frac {\sum_{i=1}^{n}x_{i}} {n}$$

mean(quakes$mag)

The Median

• Definition - The middle score when scores are ordered.

median(quakes$mag)

The Mode

• Definition - The most frequent score

getmode <- function(v) {
  uniqv <- unique(v)
  uniqv[which.max(tabulate(match(v, uniqv)))]
}

getmode(quakes$mag)

More Than One Mode

• Bimodal

  • Having two modes.

• Multimodal

  • Having several modes.

• Example of Bimodal Distribution:

knitr::include_graphics("pictures/introDA2/bimodal_distribution.png")

Which Measure of Central Tendency & When

• Please use the following summary table to know what the best measure of central tendency is with respect to the different types of variables.

knitr::include_graphics("pictures/introDA2/Capture_01.png")

Dispersion

• Definition - a measure of dispersion, is used to describe the variability in a sample or population.
• Often it is used in conjunction with a measure of central tendency, such as the mean or median, to provide an overall description of a set of data.

• A measure of spread gives us an idea of how well the central tendency represents the data.

• We will be looking at the key measures of:

  • Range,
  • Quartiles,
  • Variance, and
  • Standard deviation.

Range

• Definition - The smallest score subtracted from the largest.

  • Very biased by outliers.

• Calculate the range using the following:

range(quakes$mag)
psych::describe(quakes$mag)

Interquartile Range

• Definition - is the difference between the first and third quartiles, Q3 - Q1.

• Quartiles Definition - The three values that split the sorted data into four equal parts.

  • Lower quartile = median of lower half of the data (25%)
  • Second quartile = median (50%)
  • Upper quartile = median of upper half of the data (75%)

knitr::include_graphics("pictures/introDA2/interquartile_range.png")

Calculating Quartiles

quantile(quakes$mag)
summary(quakes$mag)

• You can specify the percentiles using a c() function to create a vector. See as followed:

quantile(quakes$mag, c(0.05,0.50,0.75,0.95))

Variance

• Definition - The variance is the "average" of the squared deviations from the mean.

$$SD^2 = \frac {\sum_{i=1}^{n}(x_{i} - \bar{x})^2} {n}$$

var(quakes$mag)

Standard Deviation

• Definition - The standard deviation is the square root of the variance.

sd(quakes$mag)

Using Standardized Values

• Definition - A standardized value, commonly called a z-score, provides a relative measure of the distance an observation is from the mean, which is independent of the units of measurement.

$$Z = \frac{(x_{i} - \bar{x})} {SD}$$

• The z-score for the $i^{th}$ observation in a data set is calculated as follows:

quakes$zscore <- scale(quakes$mag)
head(quakes$zscore)
str(quakes$zscore)
mean(quakes$mag)
sd(quakes$mag)

Z-score properties

• The numerator represents the distance that $x_i$ is from the sample mean.

• By dividing by the standard deviation, $SD$, we scale the distance from the mean to express it in units of standard deviations.

  • A negative value indicates that $x_i$ lies below the mean.
  • A positive value indicates that it lies above the mean.

Z-score properties

• Properties:

  • 1.96 cuts off the top 2.5% of the distribution.
  • (-)1.96 cuts off the bottom 2.5% of the distribution.
  • As such, 95% of z-scores lie between (-)1.96 and 1.96.
  • 99% of z-scores lie between (-)2.58 and 2.58.
  • 99.9% of them lie between (-)3.29 and 3.29.

Measures of Association

• Two variables have a strong statistical relationship with one another if they appear to move together.
• When two variables appear to be related, you might suspect a cause-and-effect relationship.
• Sometimes, however, statistical relationships exist even though a change in one variable is not caused by a change in the other.

Covariance

• Definition is a measure of the linear association between two variables, $X$ and $Y$.
• The covariance between X and Y is the average of the product of the deviations of each pair of observations from their respective means.
• This formula is for the population covariance (use N-1 for sample covariance):

$$cov_{x,y} = \frac {\sum_{i=1}^{n}(x_{i} - \bar{x})(y_{i} - \bar{y})} {n}$$

cov(quakes$mag, quakes$depth)

Correlation

• Definition - is a measure of the linear relationship between two variables, $X$ and $Y$, which does not depend on the units of measurement. - Correlation is measured by the correlation coefficient, also known as the Pearson product moment correlation coefficient.

cor(quakes$mag, quakes$depth)

Correlation

• Correlation can easily be visualized and help reveal patterns in the data.

library(corrplot)
corrplot(cor(quakes), order = "hclust")

Summary

• In this section, you learned about:

• Frequency Distributions: helping us understand shape: skew, kurtosis

• Central tendency: mean, median, mode
• Dispersion: spread of the data, variance, standard deviation, and z-scores
• Association: covariance and correlation

doomlab/learnSTATS documentation built on June 9, 2022, 12:54 a.m.