A class is a category or range of data values. We usually divide a data set into a number of classes to help us detect patterns in the data.

The number of  classes should be between 5 and 20 to make it easier to detect patterns. Use one of the following formulae to guide your selection of a number of classes to use:

for relatively small data sets: classes = √n

for large data sets: classes = 1 + 3.3log(n)

Class width is the distance from the lower limit (starting value) of one class to the lower limit of the next class.

class width = range/number of classes

You must round up to ensure you include all data values. With integer (whole number) data, you must round to the next whole number if this formula yields a whole number.

The frequency f of a class is the number of data entries in the class.

The relative frequency of a class is the class frequency divided by the sample size. (You may express it as a non-percent decimal ".355" or as a percentage "35.5%."

relative frequency = f/n

A frequency distribution is a table with one column for the classes into which a data set has been divided and another column for the frequency of each class.

A frequency histogram is a bar graph that represents the frequency distribution of a data set. A histogram has the following properties:

1. The horizontal scale (x-axis) is quantitative and measures the data values.
2. The vertical scale (y-axis) measures the frequency of the classes.
3. Consecutive bars must touch.

The cumulative frequency for a class is the sum of the frequency for that class and the frequencies of all previous classes.

A measure of central tendancy is a value that represents a typical, or central, entry of a data set.

The mean, median and mode of a data set are measures of central tendancy.

The mean of a data set is the sum of the data entries divided by the number of entries. The mean is not resistant to outlying values.

population mean: μ=Σx/N

sample mean: x bar = Σx/n

The median of a data set is the value that lies in the middle of the data when the data set is ordered.

The median is resistant to outlying values.

The mode of a data set is the data entry that occurs with the greatest frequency. If no data entry is repeated, the data set has no mode. If two entries occur with the same greatest frequency, each entry is a mode and the data set is called bimodal.

For qualitative data, mode is the only measure of central tendancy.

A frequency distribution is symmetric when a vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images.

A frequency distribution is uniform when all entries, or classes, in the distribution have equal or approximately equal frequencies. A uniform distribution is also symmetric.

A frequency distribution is skewed if the "tail" of the graph elongates more to one side than to the other.

A distribution is skewed left (negatively skewed) if the tail extends to the left. A distribution is skewed right (positively skewed) if the tail extends to the right.

 A measure of variation is a numerical description of how "spread out" a data set is.

Range, variance and standard deviation are measures of variation.

The range of a data set is the difference between the maximum and minimum data entries in the set.

range = (maximum entry) – (minimum entry)

the deviation of an entry x in a data set is the difference between the entry and the mean of the data set.

population: deviation of x = x – μ

sample: deviation of x = x – x bar

population: (x – μ)²

sample: (x – x bar)²

population: Σ(x – μ)²

sample: Σ(x – x bar)²

population: σ² = Σ(x – μ)²/N

sample: s² = Σ(x – x bar)²/(n-1)

population: σ = √(Σ(x – μ)²/N)

sample: s = √(Σ(x – x bar)²/(n-1))

A measure of position specifies where in a data set an entry is located.

Quartiles, percentiles ans z-scores are measures of position.

The five-number summary of a data set refers to the minimum entry, the first quartile (Q₁), the median (Q₂), the third quartile (Q₃), and the maximum entry.

The minimum is the smallest value in a data set.

The maximum is the largest value in a data set.

A quartile is one of three numbers (the first, second and third quartiles, Q₁, Q₂ and Q₃) that divide a data set into four approximately equal parts. 75% of the values in a data set fall below the third quartile.

A percentile is one of 99 numbers (P1 to P99) that divide a data set in 100 roughly equal parts. 99% of the data values fall below the 99th percentile.

The interquartile range (IQR) is the difference between the third and first quartiles.

IQR = Q3 – Q1

The z-score represents the number of standard deviations a given value falls from the mean.

z = (x – μ)/σ