Definitions

• Population: a particular group

• **sample: a subset of the population.

• parameters: numerical description of a population characteristic (mean height…).

• sample statistics: numeric description of particular sample characteristic.

• *Branches of statistics: 1) *Descriptive statistics 2) Inferential statistics: use descriptive statistics to estimate population parameters - infer from the sample data the population statistics.

• mean:

1) population mean: $\mu = \frac{\sum_i x_i}{N}$ 2) sample mean: $\bar{x} = \frac{\sum_i x_i}{n}$

• Median: middle value in an ordered list
• If odd samples -> median is middle
• If even samples -> average the 2 middle points

1 2 3 4 5 6 7 8 9 10 11 12 13 |

• Variance / Standard Deviation:

• population variance: $\sigma^2 = \frac{\sum_i (x_i - \mu)^2}{N}$

• sample variance: $\s^2 = \frac{\sum_i (x_i - \mu)^2}{n-1}$

• Coefficient of variation: CV is used to compare 2 dataset to know which one is more spread

$$CV = \frac{s}{\bar{x}} \times 100 [%]$$

• Standard deviation of grouped data:

We cannot use $s = \sqrt{\frac{\sum_i (x_i - \bar{x})^2}{n-1}}$ because we do not know $x_i$ nor $\bar{x}$. For grouped data, the standard deviation is given by:

$$\begin{align} s &= \sqrt{ \frac{\sum_c f_c (x_c - \bar{x})^2}{n-1} } = \sqrt{\frac{\sum_c f_c x_c ^2 + \bar{x}^2 - 2 x_c \bar{x} }{n -1} } \\ &=\sqrt{\frac{\sum_c f_c x_c ^2 + \bar{x}^2 - 2 x_c \bar{x} }{n -1} } \end{align}$$

where $f_c$ is the count/frequency, $x_c$ is the midpoint of the class, $n=\sum_c f_c$ is the sample size and $\bar{x} = \frac{\sum_c f-c x_x}{\sum_c f_c}$ is the sample mean for a frequency distribution.

Grades	Frequency	Midpoint
94-100	5	97
87-93	8	90
80-86	12	83
73-79	7	76
66 - 72	4	69