Grouped data

Here data are grouped into “classes”, which are often equal-sized intervals. Instead of seeing the original values, you would be given frequencies or number of values described by a class. Statistics such as mean and quartiles are estimates and not exact.

Overview
Classes (intervals)
Frequency tables and histograms
- mean ( $\mu$ )
- HL: variance ( $\sigma^2$ ) and standard deviation ( $\sigma$ ) by hand
Cumulative frequency diagrams
Calculator

Overview

quantity	what to use	how to estimate
mean	mid-interval values	by hand, by GDC
stdev, variance	mid-interval values	by hand (HL only), by GDC (SL and HL)
modal class	interval	by hand, by GDC (if given cumulative frequency table)
median	mid-interval values	by hand, by GDC
quartiles, IQR	upper/right endpoints	by GDC
percentiles	upper/right endpoints	from cumulative frequency diagrams

Classes (intervals)

Classes on tests will be equally spaced. For calculations, each class will be represented by its median.

For continuous data, the median is the mean between the start of this class and the start of next class.

For discrete data, the median, is the mean between the start of this class and the end of this class.

Consider the intervals

20 < x \leq 30,\;\; 30 < x \leq 40,\;\; 40 < x \leq 50

For continuous data, the mid-interval values are $25$ , $35$ , and $45$ .

For discrete data, say only integers, the mid-interval values are $\frac{21 + 30}{2} = 25.5$ , $35.5$ , and $45.5$ .

For another set of interval classes,

15 \leq x < 25,\;\; 25 \leq x < 35,\;\; 35 \leq x < 45

For continuous data, the mid-interval values are $20$ , $30$ , and $40$ .

For whole number data, the mid-interval values are $\frac{15 + 24}{2} = 19.5$ , $29.5$ and $39.5$ .

Frequency tables and histograms

The mean is estimated using the mid-interval values and frequencies. This is usually done on the calculator.

The modal class is the one with the highest frequency.

The formulas are estimates.

mean ( $\mu$ )

\bar x = \frac 1n \sum_{x} f_xx

where $f_x$ is the frequency for the corresponding min-interval values $x$ , and

n = \sum_x f_x

HL: variance ( $\sigma^2$ ) and standard deviation ( $\sigma$ ) by hand

The formula for variance, with frequencies $f$ , is

s^2 = \frac 1n \sum_{x} f_x(x - \bar x)^2 = \frac 1n \sum_{x} f_xx^2 - {\bar x}^2

Note that the $- {\bar x}^2$ is outside the summation. Taking the positive square root of both sides is the formula for standard deviation.

Cumulative frequency diagrams

In cumulative frequency diagrams, the $y$ -value is the sum of frequencies up to each $x$ -value.

When constructing cfd from frequency tables or histograms, At the upper bound of each class, the frequency is added to the existing total (cumulative) frequency. Connect the dots using straight lines.

For quartile (and median) calculations, first find the total number of frequencies (data points). This may or may not be given in the question.

From $x$ values of $25\%$ , $50\%$ , $75\%$ times the total, read off the $y$ values of $Q1$ , $Q2$ (median), and $Q3$ .

For mean, the diagram should be subdivided into equal classes and enter class mid-interval values and frequencies into calculator. The question should suggest a class size.

The x-values of the 25%, 50%, and 75% of maximum cumulative frequencies correspond to Q1, Q2, and Q3 of box and whiskers plot — Box and whiskers plot is basically cumulative frequency diagram projected down to 1-D.

Calculator

Group data are still one-variable statistics. Calculator should provide an option to enter frequencies.

Be able to calculate all the statistics as you would for a data list. Interpreting statistics on TI-84 Plus (example using discrete random variables), where frequencies are analogous to probabilities.

Previous : Data lists and frequency distributions

Next : Effects of changes on mean and standard deviation