Descriptive Statistics

• Measures of central tendency

• Measures of variability

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.
\[ \mu = \frac{1}{N}\sum_{i=1}^{N}Y_{i}\, \]

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. 1), 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, the measurement that partitions the ordered measurements into two halves.

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 such as \( n=4 \) and \( n=5 \)

The median is the middle measurement of the distribution (a color for each half of the distribution). The mean is the center of gravity, the point at which the frequency distribution would be balanced (if observations had weight).

Note: The mean and median have the same units as the variable.

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

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.

Note #3: If the frequency distribution is bell shaped, then about two-thirds (68%) of the observations will lie within one standard deviation of the mean, and 95% of the observations will lie within ~ two standard deviations of the mean.

Note #3: If the frequency distribution is bell shaped, then about two-thirds (68%) of the observations will lie within one standard deviation of the mean, and 95% of the observations will lie within ~ two standard deviations of the mean.

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.

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 (standard deviation and interquartile range) tend to give similar information when the frequency distribution is symmetric and unimodal (i.e. bell shaped).

Heuristic #2: The information starts to differ when the distribution is strongly skewed or there are extreme observations. The median and IQR tend to be better descriptors of the ‘typical’ observation and spread of the main part of the distribution. However, the mean and standard deviation still reflect the distribution of the data as a whole and have better mathematical properties.

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?”

\( ^* \) 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 ?quantile). For the HW, either version is acceptable. Default type in R is type=7.