Measures of central tendency are statistical measures that describe the center or average of a data distribution. They provide a single value that represents the most typical or central value in a dataset. The three main measures of central tendency are the mean, median, and mode.
- Mean:
- The mean, often referred to as the average, is calculated by summing up all the values in a dataset and then dividing the sum by the number of values.
- Formula: ( \bar{x} = \frac{\Sigma_{i=1}^{n} x_i}{n} )
- The mean is sensitive to extreme values (outliers) and may not be the best measure of central tendency if the dataset has outliers.
- Median:
- The median is the middle value in a dataset when it is ordered from least to greatest. If there is an even number of observations, the median is the average of the two middle values.
- To find the median, the data must be sorted.
- The median is less affected by extreme values than the mean and is often a better measure of central tendency for skewed distributions.
- Mode:
- The mode is the value that occurs most frequently in a dataset.
- A dataset may have no mode, one mode (unimodal), or multiple modes (multimodal).
- The mode is particularly useful for categorical data.
Example:
Consider the dataset: ( 5, 7, 8, 8, 10, 12, 15 )
Mean:
[ \bar{x} = \frac{5 + 7 + 8 + 8 + 10 + 12 + 15}{7} = \frac{65}{7} \approx 9.29 ]
Median:
Sort the data: ( 5, 7, 8, 8, 10, 12, 15 )
Since there is an odd number of observations, the median is the middle value, which is 8.
Mode:
The mode is 8 because it occurs more frequently than any other value in the dataset.
Each measure of central tendency has its strengths and weaknesses, and the choice of which to use depends on the nature of the data and the specific goals of the analysis. For symmetrically distributed data with no outliers, the mean, median, and mode are often very close in value. However, for skewed or asymmetric distributions, these measures may differ significantly.