What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

What is a measure of central tendency?

A single value that represents or typifies a whole data set, around which the observations tend to cluster, such as the mean, median or mode.

Write the formula for the arithmetic mean of ungrouped and grouped data.

For ungrouped data \(\bar{X}=\dfrac{\sum X}{N}\); for grouped data \(\bar{X}=\dfrac{\sum fX}{\sum f}\).

State two important properties of the arithmetic mean.

The sum of deviations of all items about the mean is always zero, \(\sum (X-\bar{X})=0\), and the mean is affected by extreme values.

When is a weighted arithmetic mean used?

When the different items do not have equal importance, each value is given a weight and the weighted mean \(\dfrac{\sum WX}{\sum W}\) is computed.

Define the median and its position.

The median is the positional value dividing an ordered distribution into two equal halves; its position is the \(\dfrac{N+1}{2}\)th item.

Write the formula for the median of a continuous series.

\(\text{Median}=L+\dfrac{\frac{N}{2}-c.f.}{f}\times h\), where \(L\) is the lower limit of the median class, \(c.f.\) the cumulative frequency before it, \(f\) its frequency and \(h\) its width.

What are quartiles and percentiles?

Quartiles \(Q_1\), \(Q_2\) and \(Q_3\) divide an ordered distribution into four equal parts, while percentiles divide it into a hundred equal parts; the 50th percentile and \(Q_2\) both equal the median.

Define mode and write its formula for a continuous series.

The mode is the most frequently occurring value; for a continuous series \(\text{Mode}=L+\dfrac{D_1}{D_1+D_2}\times h\), where \(D_1\) and \(D_2\) are the differences of the modal-class frequency from the preceding and succeeding classes.

Why is the median preferred over the mean for some data?

Because the median is a positional value not affected by extreme values, so it better represents data with very large or very small items, and it can be found for open-ended distributions.

State the relation between mean, median and mode.

The median always lies between the mean and the mode; if the mean is the largest the order is mean, median, mode, and if smallest the order reverses.

Measures of Central Tendency

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Summary

A measure of central tendency summarises a whole data set into a single representative or typical value. The three most commonly used averages are the arithmetic mean, the median and the mode (the geometric and harmonic means are used in special cases). The arithmetic mean is the sum of all observations divided by their number, \(\bar{X}=\dfrac{\sum X}{N}\); for grouped data it is \(\dfrac{\sum fX}{\sum f}\), and it can also be found by the assumed-mean method, \(\bar{X}=A+\dfrac{\sum fd}{\sum f}\), or the step-deviation method. Its two key properties are that the sum of deviations of items about the mean is always zero and that it is affected by extreme values. When items differ in importance, a weighted arithmetic mean is used. The median is the positional middle value that divides an ordered distribution into two equal halves; its position is the \(\dfrac{N+1}{2}\)th item, and for a continuous series \(\text{Median}=L+\dfrac{\frac{N}{2}-c.f.}{f}\times h\). Unlike the mean, the median is not affected by extreme values. Related positional measures are quartiles, which divide the data into four equal parts (\(Q_1\), \(Q_2=\) median, \(Q_3\)), and percentiles, which divide it into a hundred parts. The mode is the most frequently occurring value, found for grouped data from the modal class as \(\text{Mode}=L+\dfrac{D_1}{D_1+D_2}\times h\). The median always lies between the mean and the mode. The mean uses all observations but is distorted by extreme items; the median and mode suit open-ended distributions and the mode suits qualitative data.

Measures of Central Tendency

Key terms

Arithmetic mean

The sum of all observations divided by their number, \(\bar{X}=\dfrac{\sum X}{N}\); the most common average.

Weighted arithmetic mean

A mean in which each value is multiplied by a weight reflecting its importance, \(\dfrac{\sum WX}{\sum W}\).

Median

The positional middle value that divides an ordered distribution into two equal halves; not affected by extreme values.

Quartiles

Values \(Q_1\), \(Q_2\) and \(Q_3\) that divide an ordered distribution into four equal parts.

Mode

The value that occurs most frequently in a data set; used for qualitative data.

Modal class

The class with the highest frequency, used to compute the mode in a continuous series.

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A measure of central tendency is a single value that:

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Measures of Central Tendency

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