Descriptive Statistics Classrooms

86.1 Measures of Central Tendency

Measures of central tendency are single values that attempt to describe a set of data by identifying the central position within that set of data. These measures are often referred to as averages.

The three main measures of central tendency are the Mean, Median, and Mode.

86.2 Mean, Median, Mode (Discrete)

For discrete (ungrouped) data, calculating the mean, median, and mode involves straightforward steps.

• Mean: The sum of all values divided by the number of values.
• Median: The middle value in an ordered dataset. If there are two middle values, it's their average.
• Mode: The value that appears most frequently in the dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode.

86.3 Mean, Median, Mode (Grouped)

For grouped data, calculating central tendency measures requires using class midpoints and frequencies.

• Mean: Calculated using the formula $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$, where $f_i$ is the frequency of each class and $x_i$ is the midpoint of each class.
• Median: Found by first identifying the median class (the class containing the $(N/2)^{th}$ value, where $N$ is the total frequency), then using the formula $L + \frac{\frac{N}{2} - C_f}{f_m} \times w$, where $L$ is the lower boundary of the median class, $C_f$ is the cumulative frequency of the class before the median class, $f_m$ is the frequency of the median class, and $w$ is the class width.
• Mode: The midpoint of the class with the highest frequency (modal class). For a more precise mode, the formula $L + \frac{d_1}{d_1 + d_2} \times w$ can be used, where $L$ is the lower boundary of the modal class, $d_1$ is the difference between the frequency of the modal class and the frequency of the class before it, $d_2$ is the difference between the frequency of the modal class and the frequency of the class after it, and $w$ is the class width.

86.4 Standard Deviation

Standard deviation measures the average amount of variability or dispersion in a set of data. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation indicates that data points are spread out over a wider range of values.

The formula for population standard deviation is $\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}}$, and for sample standard deviation is $s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}$.

86.5 Quartiles, Deciles, Percentiles

These are measures of position that divide a dataset into equal parts.

• Quartiles: Divide a dataset into four equal parts. ($Q_1$ (25th percentile), $Q_2$ (50th percentile, also the median), $Q_3$ (75th percentile)).
• Deciles: Divide a dataset into ten equal parts.
• Percentiles: Divide a dataset into one hundred equal parts. The $P^{th}$ percentile is the value below which $P$ percent of the observations fall.