Measure of Center A measure of center is a value at the center or middle of the data set Surprising, huh?

Mean Arithmetic mean (or just mean) A measure of center found by adding the data values and dividing by the number of values This is what is usually called an average. That term is imprecise, and we should stick with mean

Notation Sigma - means “add these up” (sum) The variable used to represent the data values The size of the sample (number of data values) The size of the population (number of data values) Mean of the sample data Read: x-bar Mean of the population values (all of them) Read: Like “you” with an “m”

Example 6 students are asked how long they studied last week. Data: 7, 8, 10, 11, 13, 25 So for our sample, the mean hours studied was 12.3 hours

Note The mean can be strongly influenced by outliers, since it takes into account every value. Data: 7, 8, 10, 11, 13, 25 Mean: 12.3

Median The median is the middle value when the data is listed in order. Median is denoted If the number of data values is odd, the median is the middle data value If the number of data value is even, there is no middle data value, so we find the mean of the two numbers in the middle

Example Data: 5, 6, 8, 11, 13, 25 Even number of data values, so the median is found by finding the mean of the two middle numbers, 8 and 11. (8+11)/2 = 9.5 Data: 5, 6, 8, 11, 13, 25, 26 Odd number of data values, so the median is the middle value, 11

Mode The mode of the data is the value that occurs most often. The mode is denoted by M It is possible to have one mode, two modes (bimodal), many modes (multimodal), or no modes at all (when no data is repeated) The mode is most commonly used with data at the nominal level, since it is the only measure of center that can be done.

Midrange Midrange is a mostly useless measure of center. It is determined by finding the mean of the highest and lowest data values.

Rule for rounding Never round while doing your calculations (more than absolutely necessary) Round your final answer so that it has one more decimal place than the original data did. This is primarily so our mean doesn’t imply we know more than we actually do.

Example For the data: 5, 6, 8, 11, 13, 25 we found the mean to be 11.3 The exact value was 11.33333333333333333 If we rounded to 11.33, it would suggest that we had measured minutes, not just hours. We don’t want to make our mean appear more accurate than it is.

Skewness and the measures of center

Our Coffee Data Mean: 6.38 hrs Median: 6.85 hrs Modes: 5.3, 7.2, 8.2 Distribution: Strongly skewed to the left

Homework 2.4: 3, 9 Think about: 22, 23, 24