Descriptive Statistics Statistical Notation Measures of Central Tendency Measures of Variability Estimating Population Values

Statistical Notation Statistics for populations are called parameters and are represented with Greek letters ( ,  ). Statistics for samples are called statistics and are represented with English letters (M, SD).

Summation Notation The capital Greek letter  means “sum of”  x means add up all the scores on variable x If the scores on x are 2, 3, 5,  x = 10

Measures of Central Tendency Indicates the typical score in a distribution Mean (M,  ) is the arithmetic average Median (Mdn) is the middle score (50 th %ile) Mode is the most common score

Measures of Central Tendency The mean requires interval/ratio data; affected by all scores; can be distorted by outliers. The median requires ordinal or higher level data; not distorted by outliers. The mode can be used with any type of data; unstable in small samples.

Measures of Variability Indicate spread of scores The range is the difference between the highest and lowest scores. The standard deviation (SD,  ) is average distance from the mean. The variance (s 2,   ) is the average squared distance from the mean.

Measures of Variability The range can be unstable in small samples. Standard deviation is the most commonly used measure of variability. Variance is difficult to interpret because it is in squared units, but is used in inferential statistics. All measures require interval/ratio data.

Estimating Population Values We are sometimes interested in estimating descriptive statistics for a population based on a sample. The best estimate of the population mean is the sample mean. The sample mean is an unbiased estimate.

Estimating Population Values The best estimate of the population  is not the sample s. The sample s tends to underestimate the population  ; it is a biased estimate. The formula for estimated population standard deviation compensates for this bias.