Question

how do you find the iqr?
in what shape distributions is it better to use the mean?
draw all four shape distributions?
when is the mean > then the median?
when is the mean < then the median?
when is the mean = median?
define variability.
what do we use to describe variability in a box plot?
how can you determine if you have an outlier? what are the two rules?

Explanation:

Step1: Find IQR

The inter - quartile range (IQR) is found by subtracting the first quartile ($Q_1$) from the third quartile ($Q_3$), i.e., $IQR = Q_3 - Q_1$.

Step2: Distributions for mean

The mean is a better measure of central tendency for symmetric distributions (such as the normal distribution) as the data is evenly distributed around the center.

Step3: Draw distributions

Common distributions include symmetric (bell - shaped like normal), right - skewed (tail on the right), left - skewed (tail on the left), and uniform (rectangular - shaped). However, drawing is beyond text - based response. In a right - skewed distribution, the mean is pulled in the direction of the tail, so mean > median. In a left - skewed distribution, the mean is pulled in the direction of the tail, so mean < median. In a symmetric distribution, mean = median.

Step4: Define variability

Variability refers to how spread out a set of data is. It can be measured in different ways such as range, variance, standard deviation, and IQR.

Step5: Variability in box plot

In a box plot, the IQR (the length of the box) is used to describe the variability of the middle 50% of the data.

Step6: Determine outliers

The two common rules for determining outliers are: 1. The 1.5 * IQR rule: Any value less than $Q_1-1.5\times IQR$ or greater than $Q_3 + 1.5\times IQR$ is considered an outlier. 2. The 3 - standard - deviation rule (for normal distributions): Values more than 3 standard deviations away from the mean are considered outliers.

Answer:

1. $IQR = Q_3 - Q_1$
2. Symmetric distributions
3. Cannot draw in text, but types are symmetric, right - skewed, left - skewed, uniform
4. In right - skewed distributions, mean > median
5. In left - skewed distributions, mean < median
6. In symmetric distributions, mean = median
7. Variability is how spread out data is
8. IQR
9. 1. $Q_1-1.5\times IQR$ and $Q_3 + 1.5\times IQR$ rule; 2. 3 - standard - deviation rule (for normal distributions)