Data Analysis: Practice Problems and Methods

Averages, quartiles and box plots

The median is the middle value of the sorted data. Split the data at the median into a lower and an upper half (leaving out the median when the count is odd); the median of the lower half is the first quartile $Q_1$ and that of the upper half the third quartile $Q_3$.

The interquartile range is $Q_3 - Q_1$. A box plot shows the five values: minimum, $Q_1$, median, $Q_3$ and maximum.

Variance and standard deviation

The variance is the mean of the squared deviations, or equivalently "mean of squares minus square of the mean". If $y = ax + b$, then $\bar{y} = a\bar{x} + b$, $s_y^2 = a^2 s_x^2$ and $s_y = |a|s_x$.

Covariance and correlation

The closer $r$ is to 1, the stronger the positive correlation; the closer to $-1$, the stronger the negative correlation. Under $u = ax + b$, $v = cy + d$, the covariance is multiplied by $ac$ and the correlation coefficient changes only by the sign of $ac$.

In hypothesis testing, if the probability of the observed result under a hypothesis is below a chosen level (for example 5%), we conclude that the hypothesis was wrong.