- Parameter: A numerical characteristic that describes a population (e.g., population mean $\mu$, population standard deviation $\sigma$). Parameters are usually unknown and are estimated from sample statistics.

- Statistic: A numerical characteristic that describes a sample (e.g., sample mean $\bar{x}$, sample standard deviation $s$). Statistics are calculated from sample data and are used to estimate population parameters.

- Systematic Sampling: Selecting every $k^{th}$ element from an ordered list of the population, after a random start.

- Relative Frequency: The proportion of observations in a class interval, calculated as $\text{Frequency} / \text{Total Number of Observations}$.

- Sample Mean: $\bar{x} = \frac{\sum x}{n}$

where $\sum x$ is the sum of all values in the sample, and $n$ is the sample size.

- Population Mean: $\mu = \frac{\sum x}{N}$

where $\sum x$ is the sum of all values in the population, and $N$ is the population size.

- If $n$ is odd, the median is the value at the $\left(\frac{n+1}{2}\right)^{th}$ position.

- If $n$ is even, the median is the average of the two middle values at the $\left(\frac{n}{2}\right)^{th}$ and $\left(\frac{n}{2}+1\right)^{th}$ positions.

- $\text{Range} = \text{Maximum Value} - \text{Minimum Value}$

- Sample Variance: $s^2 = \frac{\sum (x - \bar{x})^2}{n-1}$

(using $n-1$ for unbiased estimation of population variance)

- Population Variance: $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$

- Sample Standard Deviation: $s = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}$

- Population Standard Deviation: $\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$

- $CV = \frac{s}{\bar{x}} \times 100\%$ (for sample) or $CV = \frac{\sigma}{\mu} \times 100\%$ (for population)

- $IQR = Q_3 - Q_1$

- $Q_1$ (First Quartile) = $25^{th}$ percentile

- $Q_2$ (Second Quartile) = $50^{th}$ percentile = Median

- $Q_3$ (Third Quartile) = $75^{th}$ percentile

- $z = \frac{x - \mu}{\sigma}$ (for population)

- $z = \frac{x - \bar{x}}{s}$ (for sample)

* Parameter: A numerical characteristic of a population (e.g., $\mu$, $\sigma$).

* Statistic: A numerical characteristic of a sample (e.g., $\bar{x}$, $s$).

* Mean: The arithmetic average of a dataset ($\bar{x}$ or $\mu$).

* Variance: The average of the squared differences from the mean ($s^2$ or $\sigma^2$).

* Standard Deviation: The square root of the variance, in the same units as the data ($s$ or $\sigma$).

* Coefficient of Variation (CV): A measure of relative variability, $\frac{s}{\bar{x}} \times 100\%$.

* Interquartile Range (IQR): The range of the middle 50% of the data ($Q_3 - Q_1$).

* Z-score: A measure of how many standard deviations a data point is from the mean ($z = \frac{x - \mu}{\sigma}$).

* Mean: $\bar{x} = \frac{75+80+65+90+75+85+70}{7} = \frac{540}{7} \approx 77.14$

* Range: $90 - 65 = 25$.

* Sample Variance: $s^2 = \frac{(75-77.14)^2 + (80-77.14)^2 + \dots + (70-77.14)^2}{7-1} \approx 78.57$

* Sample Standard Deviation: $s = \sqrt{78.57} \approx 8.86$

* Example: A student scores 85 on a test where the class mean ($\mu$) was 70 and the standard deviation ($\sigma$) was 10.

* $z = \frac{85 - 70}{10} = \frac{15}{10} = 1.5$

* Explanation: The student's score of 85 is 1.5 standard deviations above the class average. This indicates a relatively strong performance compared to the rest of the class. If another student scored 60 on a different test with $\mu=50, \sigma=5$, their z-score would be $z = \frac{60-50}{5} = 2.0$, indicating a relatively even stronger performance on their respective test.

The heart of descriptive statistics lies in its measures of central tendency and measures of dispersion. For central tendency, the document thoroughly explains the mean ($\bar{x} = \frac{\sum x}{n}$ for sample, $\mu = \frac{\sum x}{N}$ for population), median (the middle value of an ordered dataset), and mode (the most frequent value). It discusses their properties, sensitivities to outliers, and appropriate use cases. For dispersion, measures like range (Max - Min), variance ($s^2 = \frac{\sum (x - \bar{x})^2}{n-1}$ for sample, $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$ for population), and standard deviation ($s = \sqrt{s^2}$, $\sigma = \sqrt{\sigma^2}$) are meticulously defined and explained. The coefficient of variation ($CV = \frac{s}{\bar{x}} \times 100\%$) is introduced as a measure of relative variability, useful for comparing datasets with different scales. The interquartile range (IQR) ($Q_3 - Q_1$) is also covered as a robust measure of spread.

Finally, the document delves into measures of position, which describe the relative standing of individual data points. This includes percentiles and quartiles ($Q_1, Q_2, Q_3$), which divide data into equal parts. A particularly important concept is the Z-score ($z = \frac{x - \mu}{\sigma}$), which quantifies how many standard deviations a data point is from the mean, allowing for standardized comparisons across different datasets. The utility of box-and-whisker plots in visually summarizing these positional measures is also highlighted.