Descriptive Statistics

• Homework #3: Stay tuned (announcing today)
• Reading Assignment for Friday - W&S, Chapter 4

Definition:The frequency distribution of a variable is the number of occurrences of all values of that variable in the data.

Definition:The relative frequency distribution of a variable is the fraction of occurrences of all values of that variable in the data or population.

• These definitions apply to both continuous and discrete variables.
• Frequency = Number
• Relative frequency = Fraction (proportion)
• For data, the relative frequency distribution (in the form of a histogram or bar plot) is an estimate of the population relative frequency distribution.

Question:What type of plot represents the frequency (relative frequency) distribution for a discrete variable?

Answer:Bar plot

Definition: A bar plot uses the height of rectangular bars to display the frequency distribution (or relative frequency distribution) of a categorical variable.

• i.e. height of bars = number or proportion

Question: What type of plot represents the frequency distribution for a continuous variable?

Answer: Histogram

Definition: A histogram for a frequency distribution uses the height of rectangular bars to display the frequency distribution of a numerical variable.

Definition: A histogram for a relative frequency distribution uses the area of rectangular bars to display the frequency distribution of a numerical variable.

Load and show the data:

salmonSizeData <- read.csv(url("http://whitlockschluter.zoology.ubc.ca/wp-content/data/chapter02/chap02f2_5SalmonBodySize.csv"))
head(salmonSizeData)

  year   sex oceanAgeYears lengthMm massKg
1 1996 FALSE             3      513  3.090
2 1996 FALSE             3      513  2.909
3 1996 FALSE             3      525  3.056
4 1996 FALSE             3      501  2.690
5 1996 FALSE             3      513  2.876
6 1996 FALSE             3      501  2.978

Definition: The population mean \( \mu \) is the sum of all the observations in the population divided by \( N \), the number of observations in the population (assuming it is finite - for now).
\[ \mu = \frac{1}{N}\sum_{i=1}^{N}Y_{i}\, \]

Definition: The sample mean \( \overline{Y} \) is the sum of all the observations in the sample divided by \( n \), the number of sample observations.
\[ \overline{Y} = \frac{1}{n}\sum_{i=1}^{n}Y_{i}\, \]

Question: Is the population mean \( \mu \) a parameter or an estimate? What about the sample mean?

Note that every observation has equal weight (i.e. \( \frac{1}{n} \)), so any outliers can strongly affect the mean. It is a very democratic statistic - equal representation!

Definition: The population median is the middle measurement of the set of all observations in the population (again, assume population finite for now).

Definition: The sample median is the middle measurement of the set of all observations in the sample.

How do you compute the median? W&S version:

• First, sort the data from smallest to largest.
• We then have two conditions:
  • If the number of observations is odd, then we have \[ Median = Y_{(n+1)/2} \]
  • If the number of observations is even, then we have \[ Median = \left[Y_{n/2} + Y_{(n/2)+1}\right]/2 \]

Look at special cases of \( n=3 \) and \( n=4 \)!!!

Definition: The population variance \( \sigma^{2} \) is the average of the squared deviations of all observations from the population mean, and assuming a finite population, we have
\[ \sigma^{2} = \frac{1}{N}\sum_{i=1}^{N}(Y_{i}-\mu)^2 \]

Definition: The sample variance \( s^{2} \) is the average of the squared deviations from the sample mean,
\[ s^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-\overline{Y})^2 \]

Question: Why \( n-1 \)??

Answer: Needed to be unbiased estimate!!

Definition: The population standard deviation \( \sigma \) is the square root of population variance
\[ \sigma = \sqrt{\sigma^{2}} \]

Definition: The sample standard deviation \( s \) is the square root of the sample variance,
\[ s = \sqrt{s^{2}} \]

Note #1: \( s \) is in general a biased estimator of \( \sigma \). The bias gets smaller as the sample size gets larger.

Note #2: \( s \) and \( \sigma \) have the same units as the random variable!!!

Definition: The interquartile range \( IQR \) is the difference between the third and first quartiles of the data. It is the span of the middle 50% of the data.

Heuristic #1: The locate (mean and median) and spread (interquartile range and standard deviation) give similar information when the frequency distribution is symmetric and unimodal (i.e. bell shaped).

Heuristic #2: The mean and standard deviation become less informative when the distribution is strongly skewed or there there are extreme observations.

Since in biology many times the standard deviation scales with the mean, it can be more informative to look at the coefficient of variation.

Definition: The coefficient of variation (CV) calculates the standard deviation as a percentage of the mean: \[ CV = \frac{s}{\bar{Y}}\times 100\% \]

In other words, the CV answers the question “How much variation is there relative to the mean?”

Make sure you read the book for the following discussions

• How to compute a mean and standard deviation from a frequency table

Question: Why is this important to know?

• Rounding rules for displaying tables and statistics
• Effect of changing measurement scale
• Cumulative frequency distributions (we will cover this later as well)

My point here is that you are responsible for all book material, even if we don't cover it in lecture!

\( ^* \) Note that IQR has different algorithms. To match the algorithm in W&S, you should use IQR(___, type=2). There are different algorithms as there are different ways to calculate quantiles. (for curious souls, see ?quantiles). For the HW, either version is acceptable. Default type in R is type=7.