Symbol Description It would be a good idea now to start looking at the symbols which will be part of your study of statistics.  The uppercase Greek letter sigma; indicates a summation of values X A variable that represents quantitative data N Number of entries in a population  The lowercase Greek letter mu; the population mean x Read as “x bar;” the sample mean

Measures of Central Tendency Mean: The sum of all data values divided by the number of values For a population: For a sample: Median: The point at which an equal number of values fall above and fall below There are other measures of central tendency (ex. midrange), but these three are the most commonly used. Be sure to discuss the advantages and disadvantages for each. Discuss the sigma notation. Mode: The value with the highest frequency

Calculate the mean, the median, and the mode An instructor recorded the average number of absences for his students in one semester. For a random sample the data are: 2 4 2 0 40 2 4 3 6 Calculate the mean, the median, and the mode Mean: Median: Sort data in order As students which measure is the most representative of the center of the data. 0 2 2 2 3 4 4 6 40 The middle value is 3, so the median is 3. Mode: The mode is 2 since it occurs the most times.

Calculate the mean, the median, and the mode. Suppose the student with 40 absences is dropped from the course. Calculate the mean, median and mode of the remaining values. Compare the effect of the change to each type of average. 2 4 2 0 2 4 3 6 Calculate the mean, the median, and the mode. Mean: Median: Sort data in order. Ask students which measure was changed the most by eliminating the extreme value (outlier). 0 2 2 2 3 4 4 6 The middle values are 2 and 3, so the median is 2.5. Mode: The mode is 2 since it occurs the most times.

Weighted Mean and Mean of Grouped Data Sometimes data sets contain entries that have a greater effect on the mean than do other entries. To find the mean of such data sets, you must find the weighted mean. The weighted mean is the mean of a data set whose entries have varying weights. A weighted mean is given by the equation to the left where w is the weight of each entry x.

The mean of a frequency distribution for a sample is approximated by: Mean of Grouped Data The mean of a frequency distribution for a sample is approximated by: Where x and f are the midpoints and frequencies of a class, respectively.

Finding the mean of a frequency distribution Find the midpoint of each class. Find the sum of the products of the midpoints and the frequencies. Find the sum of the frequencies Find the mean of the frequency distribution.

Shapes of Distributions Symmetric Uniform Mean = Median Skewed right Skewed left Both curves at the top are symmetric. Note that the fulcrum is placed at the mean of each distribution. The direction of skew is with the tail. The mean is always in the direction of the skew. Mean > Median Mean < Median