Summarize Numerical Data

Measure Centre

Mean, median, and mode are three kind presentation of the “centre”. 

Mean

Mean: is the “average” you’re used to, where you add up all the numbers and then divide by the number of numbers.

Ｍedian

Median is the “middle” value in the list of numbers. To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first.

Mode

Mode is the value that occurs most often. If no number is repeated, then there is no mode for the list.

An example

Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

Mean: (13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15

Median: 13, 13, 13, 13, 14, 14, 16, 18, 21

Mode: the occur of 13 is more than other numbers. So 13 is the mode.

• mean: 15
• median: 14
• mode: 13

Measure Spread

Interquartile range (IQR), Range and Standard Deviation are two summarise of data spread.

Interquartile range (IQR)

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles.

IQR also relies on Q1, Q2, and Q3, respectively.

• Q1 is the “middle” value in the first half of the rank-ordered data set.
• Q2 is the median value in the set.
• Q3 is the “middle” value in the second half of the rank-ordered data set.

IQR=Q3-Q1

Range

Range is the difference between the largest value and smallest value.

Standard Deviation

The Standard Deviation is a measure of how spread out numbers are. Its symbol is σ (the greek letter sigma)

The formula is easy: it is the square root of the Variance. 

Population Standard Deviation:  

Sample Standard Deviation:

Variance

The Variance is defined as: the average of the squared differences from the Mean.

An example

5, 8, 4, 4, 6, 3, 8

Put them in order: 3, 4, 4, 5, 6, 8, 8

Cut the list into quarters:

• Quartile 1 (Q1) = 4
• Quartile 2 (Q2), which is also the Median, = 5
• Quartile 3 (Q3) = 8
• IQR= Q3-Q1= 8-4 =4
• Range = 8 -3 = 5
• Standard Deviation =1.988

Summarize Categorical Data

Proportions/ Percentages

Population Distribution

Shape of Distribution

Normal Distribution

Other Population Distribution