2A. Measures of Location

Measures of Central Tendency

Measures of central tendency describes where the center of the data is located. Different measures are used in different situations based on which best describes the value you're trying to find.

Mode or modal class is the value that occurs most often this is used with both qualitative or quantitative data with one or two modes (bimodal). However it is not informative if the values only appear once.

Median is the middle value of an ordered data set. This is used with quantitative data, and often when there are extreme values as the median is unaffected.

Mean is a centre point of the data, measured by the sum of all data, divided by number of datapoints. As the arithmetic average of all values it gives a true measure of data. However, it is affected by extreme values.

The mean can be measured as follows

\overset{―}{x} = \frac{\sum x}{n}

where:
$\bar{x} =$ the mean
$\sum x =$ the sum of all data values
$n =$ the number of data values

For data in a cumulative frequency table, the mean is calculated as follows

\overset{―}{x} = \frac{\sum x f}{\sum f}

where
$\sum x f =$ sum of the products of the data values and their frequencies
$\sum f =$ the sum of frequencies

Other Measures of Location

The median $(Q_{2})$ splits the data into two equal halves, each comprising 50% of the data set.
The lower quartile $(Q_{1})$ is one quarter of the way through the dataset.
The upper quartile $(Q_{3})$ is three quarters of the way through the dataset.

Percentiles split the dataset into 100 parts, for example the $10 t h$ percentile is a tenth of the way through the data set. Meaning 10% of the dataset are lesser than the $10 t h$ percentile and 90% are greater.

To find upper and lower quartiles for discrete data

Divide $n$ by 4 (lower quartile) or find $\frac{3}{4}$ of $n$ (upper quartile)
If this is a whole number, the lower or higher quartile is the midpoint between this and the datapoint above. If it is not, round up and pick this number.

When data is in a grouped frequency table you can use interpolation to estimate the means, quartiles, and percentiles. This method assumes that data values are equally distributed across each class.

Q_{1} = \frac{n}{4} t h d a t a v a l u e

Q_{2} = \frac{n}{2} t h d a t a v a l u e

Q_{3} = \frac{3 n}{4} t h d a t a v a l u e

Source

Mogull, R. G. (1990). Popular Measures of Central Tendency. The Mathematics Teacher, 83(9), 744–746. Retrieved from https://pmt.physicsandmathstutor.com/download/Maths/A-level/Statistics/Data-Presentation-and-Interpretation-1/Cheat-Sheets/Measures of Location and Spread.pdf

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