Calculate measures of central tendency

The learner calculates and interprets the arithmetic mean, median, and mode for numerical data presented as lists or frequency tables, including using frequencies to determine totals, counts, and positions. They understand the mean as an equal-share value, the median as the middle ordered position, and the mode as the most frequent value, and can explain how skewness and outliers may cause these measures to differ. The scope excludes geometric or harmonic means and advanced statistical estimators.

Detailed Explanation: Calculate measures of central tendency

Central tendency describes the typical or central value in a data set:

Mean: the equal-share value; add all values and divide by the number of values.
Median: the middle value when the data are in order.
Mode: the value that occurs most often.

Example

The table shows the number of books read by students in a month.

Books read, $x$	Frequency, $f$	$xf$	Cumulative frequency
$2$	$1$	$2$	$1$
$4$	$1$	$4$	$2$
$5$	$2$	$10$	$4$
$6$	$1$	$6$	$5$
$7$	$1$	$7$	$6$
$20$	$1$	$20$	$7$
Total	$\mathbf{7}$	$\mathbf{49}$

1. Calculate the mean

For a frequency table, multiply each value by its frequency and add the products:

\text{Mean}=\frac{\sum xf}{\sum f}

Therefore,

\text{Mean}=\frac{49}{7}=7

The mean is $7$ books. This means that if the total of $49$ books were shared equally among $7$ students, each student would have read $7$ books.

2. Calculate the median

There are $7$ data values, so the middle position is

\frac{7+1}{2}=4

Use the cumulative frequency column to find the 4th value. The cumulative frequency reaches $4$ at $x=5$ .

Therefore, the median is

\boxed{5}

3. Calculate the mode

The mode is the value with the greatest frequency. The value $5$ occurs twice, while every other value occurs once.

Therefore, the mode is

\boxed{5}

Interpret the results

For this data set:

Mean: $7$ books
Median: $5$ books
Mode: $5$ books

The value $20$ is much larger than the other values, so it is an outlier. It pulls the mean upward from $5$ to $7$ . The median and mode are less affected by this outlier, so $5$ may better represent the typical number of books read.

Learn by doing: Calculate measures of central tendency

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Graphing - Tally Chart - Average of Series

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