Measures of Centre, Spread and Data Transformations

**Measure** | **Interpretation or calculation** Mean | x= _i f_i x_i _i f_i; affected by extreme observations. Median | Middle of the ordered data; useful when extremes distort the mean. Mode | Most frequent value; there may be several modes or none. Variance and standard deviation | Use one-variable statistics with frequencies. Variance is the square of standard deviation. Unless stated otherwise, treat the data as the population. Quartiles | Use the required calculator convention. Hand and calculator conventions can give different quartiles for a discrete data set. Example: for 2,4,4,6, the mean and median are 4, the mode is 4, the population variance is 2, and the population standard deviation is 2. **Every value becomes** | **Mean becomes** | **Standard deviation becomes** | **Variance becomes** x+b | x+b | sigma | sigma2 ax+b | a x+b | |a|sigma | a2sigma2 Changing every value is different from correcting one observation. For a negative scale factor the order reverses, although range and IQR still scale by its absolute value.