S1 Representing data

Skewness and choice of data analysis

Skewness

                          

The first distribution shown has a positive skew. This means that it has a long tail in the positive direction.

The distribution below it has a negative skew since it has a long tail in the negative direction.

Finally, the third distribution is symmetric and has no skew.

Distributions with positive skew are sometimes called "skewed to the right" whereas distributions with negative skew are called "skewed to the left."

Skewness – visuals and calculations

Calculate Q1, Q2, Q3, mode, mean and standard deviation Draw all 3 boxplots on one piece of graph paper

Data set 1 1, 3, 5, 5, 5, 7, 10Data set 2 2, 7, 7, 8, 12, 14, 20Data set 3 3, 6, 7, 9, 10, 10, 11

•For each data set find a relationship between the mode, median and mean using =,>,< symbols

•For each data set find a relationship between Q2-Q1 and Q3-Q2

•Work out 3(mean-median) standard deviation

Skewness – Using the Quartiles

                          

Q2-Q1 = Q3-Q2 Q2-Q1 < Q3-Q2 Q2-Q1 > Q3-Q2

Skewness – Using mode, median, mean

                          

Q2-Q1 = Q3-Q2 Q2-Q1 < Q3-Q2 Q2-Q1 > Q3-Q2

Mode=median=mean Mode<median<mean Mode>median>mean

Skewness calculations

You can calculate3(mean-median)Standard deviation

This gives you a value to tell you how skewed the data are.

The closer the number to zero the more symmetrical the data

Negative value means the data has a negative skew and vice versa

Comparing data setsYou should always compare data sets using •a measure of location (mean, median, mode)•a measure of spread (range, IQR, standard deviation)•skewness

•Range gives a rough idea of spread, but is affected by extreme values.•Generally only used with small data groups

•IQR not affected by extreme values•Tells you the spread of middle 50%•Often used in conjunction with median

•Mean and standard deviation generally used when data are fairly symmetrical•data size is reasonably large