Common measures of variability, skewness, and percentiles

What does the spread (variability) tell us and when can it be good or bad

The spread (variability) tells how much sample values depart from expectations and about the diversity of values within our sample.

good : job interview candidates as this maximises the number of possibilities for an employer

bad : manufacturing process where uniform production of goods is essential

Name different measures of variability

• range

• upper and lower quartiles

• interquartile range

• standard deviation

• variance

What is range

The range is the most simple measure of variability and is simply the largest value in a data set minus the smallest value.

What are the quartiles, their position in the data , and how do you calculate the quartiles

• The first quartile is Q1 and it seperates the first 25% of the data from the rest

  • n + 1 / 4 position in the data

• The second quartile is Q2 and is also the median, it seperates the first 50% of the data from the rest

• The third quartile is Q3 and it seperates the first 75% of the data from the rest

  • 3(n+1) / 4 position in the data

• Q1 or Q3 = lower value + (remainder x (upper value - lower value ))

How do you find the upper and lower quartile?

1. put the data in ascending order

2. find the position of the data

3. use formula : lower value + (remainder x (upper value-lower value))

What is the IQR

IQR= Q3-Q1

What is standard deviation

The sample standard deviation is a measure of variability which uses all the data in its calculation. The sample standard deviation measures the variability about the mean value. The sample standard deviation is denoted by s.

What is variance

The sample variance, s² is the standard deviation squared i.e.

What is skewness used for

In statistical analysis, it is not appropriate to quote all summary statistics, you need to be able to decide which ones. This is determined by the skewness of the data.

What data do we quote if a histogram is symmetrical

the mean and standard deviation

What data do we quote if a histogram is not symmetrical

the median and the IQR

What is skewness statistic

The skewness statistic measures how skewed a data set is. We use R to calculate it.

• If -0.5 ≤ skewness statistic ≤ 0.5 , the data is roughly symmetrical

• If the skewness statistic > 0.5 , the data is right tail skewed

• If the skewness < -0.5, the data is left tail skewed

How does symmetrical data relate to the mean and median

Generally, if the data is symmetrical, then the mean and median will be roughly the same

How does left tail skewed data relate to the mean and median

The mean will often be less than the median

How does right tail skewed data relate to the mean and median

The mean will often be greater than the median

What is a percentile

A percentile is a measure indicating the value below which a given percentage of observations in a group of observations falls. For example, the 20th percentile is the value below which 20 % of the observations may be found.

how to calculate a percentile

to calculate the k th percentile :

1. order data in ascending numerical data

2. Pk is located at the k/100 × (n+1) position in the data.

3. Pk = lower value+(remainder × (upper value-lower value)).