Describing and comparing distributions

Shapes of distributions

• right or left-tailed, if there is only one tail

• approx. symmetrical

• skewed to the left or right (mean is off center and the tail on one side is longer than the other, eg left tail is longer its skewed to the left)

Clusters, gaps, peaks & outliers

• cluster, groups of data

• outlier, a data point which is far of the rest

box-plot

• example: 1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 10, 11, 14, 15, 20, 21

• median would be 6

• 2, 3 is the median in the lower end and 11, 14 is the median in the upper end to build our box ends

• and the whiskers are 1 and 21

Arithmetic Mean

The "average" number; found by adding all data points and dividing by the number of data points.

example: 4 3 1 6 1 7 => $\frac {4+3+1+6+1+7}{6}=\frac{22}{6}=3.\bar{66}$

mean = $\frac{\sum x_i}{n}$ or sum of data divided by number of data points

Median

The middle number; found by ordering all data points and picking out the one in the middle (or if there are two middle numbers, taking the mean of those two numbers).

example: reorder number set to 1 1 3 4 6 7 and add the two center numbers together and halve them => 3.5

Mode

The most frequent number—that is, the number that occurs the highest number of times. In the example above the most common number is 1.

Interquartile Range (IQR

The IQR describes the middle 50% of values when ordered from lowest to highest. To find the interquartile range (IQR), ​first find the median (middle value) of the lower and upper half of the data. These values are quartile 1 (Q1) and quartile 3 (Q3). The IQR is the difference between Q3 and Q1.

Example: 4, 4, 6, 7, 10, 11, 12, 14, 15

(4+6)/2 = 5 and (12+14)/2 = 13 --> IQR = 13 - 5 = 8

Measure of spread: range, variance & standard deviation

-10, 0, 10, 20 ,30 --> mean == 10

Range is max minus min number: 30 - (-10) = 40

8, 9, 10, 11, 12 --> mean == 10, range == 4

Variance = $\bf \sigma^2$

(min value - mean)squared

-10, 0, 10, 20 ,30 (mean = 10) ==>

$\frac{(-10 -10)^2 + (0-10)^2+(10-10)^2+(20-10)^2+(30-10)^2}{5}=200$

$\sigma^2 =\frac {\sum_{i=1}^N(x_i-\mu)^2}{N}$ i=1 stands for the first value given, N the last number in list. $\mu$(mu) stand for mean.

Standard Deviation ==> sqrt of Variance ==> $\bf \sqrt{\sigma^2} == \sigma$

$\sigma = \sqrt{200} = 10\sqrt{2}$ means that is has 10 times the standard deviation.

Or we can write is as

$SD =\sqrt{\frac{\sum\vert{x}-\mu\vert{^2}}{N}}$ Standard Deviation

where $\sum$means "sum of", xxx is a value in the data set, $\mu$ is the mean of the data set, and $N$ is the number of data points in the population.

1. Find the mean

2. For each data point, find the square of its distance to the mean.

3. Sum the values from Step 2

4. Divide by the number of data points

5. Take the square root

$MAD=\frac{\sum{\vert{x}-\bar{x}\vert^2}}{n}$ Mean Absolute Deviation

What are the similarities between SD and MAD?

The formulas are very similar! They are both based on the distance from each data point to the mean $\vert{x} - \bar{x} \vert$, and they both include dividing by the number of data points $n$.

What are the differences?

The difference between the two formulas is that when calculating standard deviation, we square the distance from each data point to the mean, and we take the square root as the last step of the formula.

Which one is better?

Standard deviation is more complicated, but it has some nice properties that make it statisticians' preferred measure of spread.

Sample Standard Deviation ==> $s_{n-1}^2$or $\frac {\sum_{i=1}^N(x_i-\bar{x})^2}{n-1}$ , where $\bar{x}$is just a different way to write $\mu$. Pronounced as "sample mean". (--> un-bias sample variance)

Population and sample standard deviation

Standard deviation measures the spread of a data distribution. It measures the typical distance between each data point and the mean.The formula we use for standard deviation depends on whether the data is being considered a population of its own, or the data is a sample representing a larger population.

• If the data is being considered a population on its own, we divide by the number of data points, $N$.
• If the data is a sample from a larger population, we divide by one fewer than the number of data points in the sample, $n-1$.

Population Standard Deviation

$\sigma = \sqrt{\frac{\sum(x_i - \mu)^2} {N}}$

Sample Standard Deviation

$\sigma = \sqrt{\frac{\sum(x_i - \bar{x})^2} {n - 1}}$

Mean -> for Population we are taking a parameter, for Sample it is a statistic

	Population (parameter)	Sample (statistic)
Mean	$\mu = \frac{\sum_{i=1}^{N} x_i}{N}$	$\bar{x} = \frac{\sum_{i=1}^{n}n_i}{n}$
Variance	$\sigma^2 = \frac{\sum_{i=1}^{N}(x_i-\mu)^2}{N}$	$s_n^2=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n}$ (biased)
		$s_{n-1}^2=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}$(unbiased)