Measures of centre and grouped data

Mean, median and mode from lists, stem-and-leaf plots and frequency tables, grouping data into class intervals, histograms and estimating the mean from class centres.

From a frequency table

Score $x$	$1$	$2$	$3$	$4$
Frequency $f$	$3$	$5$	$8$	$4$

Example 1●●

Find the mean, median and mode.

$n = 20$ and $\sum fx = 3 + 10 + 24 + 16 = 53$, so the mean $= 53 \div 20 = 2.65$.

Median: the $10$th and $11$th scores. Cumulative frequencies $3, 8, 16$, so both are $3$. Median $= 3$.

Mode $= 3$ (highest frequency).

Grouped data

Large data sets are grouped into classes such as $10$–$19$. The class centre (midpoint) stands in for every value in the class, so a mean from grouped data is an estimate.

Flashcards

Mean from a table

$\dfrac{\sum fx}{\sum f}$

Class centre of $10$–$19$

$14.5$

Median position

$\dfrac{n + 1}{2}$th value

Practice

Scores $1, 2, 3, 4$ with frequencies $3, 5, 8, 4$. Mean?
Same table. Median?
Class centre of $10$–$19$?
Stem-and-leaf: stem $1$ | $2\ 5\ 8$; stem $2$ | $1\ 1\ 4$; stem $3$ | $0$. Median?
Same data. Range?
Same table as Q1. How many scores are there?
Classes $0$–$9$ ($f = 2$), $10$–$19$ ($f = 3$), $20$–$29$ ($f = 4$), $30$–$39$ ($f = 1$). Estimate the mean using the class centres.
The mean from grouped data is: