3.1 Measures of Central Tendency: Mode, Median, and Mean CHAPTER 3 AVERAGES AND VARIATION
Page 82 Averages An average is a single number used to describe an entire sample or an entire population. There are many ways to compute averages, but we will study only three of the major ones. Mode, Median, and Mean
Page 82 Mode The mode of a data set is the value that occurs most frequently. The easiest average to compute For larger data sets, it is useful to order—or sort—the data before scanning them for the mode. Not every data set has a mode The mode is not very stable. Changing just one number in a data set can change the mode dramatically. The mode is a useful average when we want to know the most frequently occurring data value
Page 83 Median The median is the central value of an ordered distribution. The median uses the position rather than the specific value of each data entry. If the extreme values of a data set change, the median usually does not change. Advantage: if a data set contains an outlier, the median may be a better indicator of the center Disadvantage: not sensitive to the specific size of a data value
Median To find the median: 1. Order the data from smallest to largest 2. For an odd number of data values in the distribution: 3. For an even number of data values in the distribution: To find the middle of a large data set:
Page 85 Mean The mean is an average that uses the exact value of each entry is the mean (sometimes called the arithmetic mean). The most important/frequently used ―average‖ When the sample size is large and does not include outliers, the mean score usually provides a better measure of central tendency. Disadvantage: it can be affected by exceptional values/outliers
Mean To compute the mean: add the values of all the entries and then divide by the number of entries.
Note AP Formula Sheet! Mean ∑ is the Greek Notation letter S called Let x represent any value in the data set ―sigma‖ The summation symbol ∑ means ―sum the following‖ ∑x (read ―the sum of all given x values‖) means add up all the x valuesThe mean of a sample is called ―x bar‖ The mean of a population is called ―mu‖n = number of data values in the sample N = number of data values in the sample μ is the Greek letter mu called ―mew‖
Using the Calculator The TI-83 and TI-84 make it fairly easy to find the mode, median, and mean. 1. Enter the data values in L1 2. Hit STAT, choose 2:SortA to organize the data 3. View the list to find the mode 4. Hit STAT, tab over to CALC, choose 1:1 – Var Stats, Hit 2nd 1 ENTERBecause the formulas forthe sample mean and thepopulation mean are thesame, this feature is usedto find, μ. It is YOUR jobto know if you areworking with a sample ora population.
Mean and Median The mean and median of two key measures of center. Important for comparing data sets State what the values are AND what they tell us about each set or one set compared to the other Important for describing data sets Gives a ―typical‖ number to consider when looking at any data sets For Example: Consider two data sets A and B where they are both roughly normal (bell-shaped) and have about the same spread. The mean of A is 6 and the mean of B is 4. You might say: ―Set A is centered at 6, where set B is centered at 4. Since the mean of set B is lower than set A, set B would have lower overall values, on average than set A‖
Page 86 Resistant Measures A resistant measure is one that is not influenced by extremely high or low data values. The mean is not a resistant measure of center because changing the size of only one data value will affect the mean and it is affected by outliers. The mean is an example of a non-resistant measure. The median is more resistant to changes in individual data values and outliers.
Page 87 Data Types and Averages Levels of Measurement: nominal, ordinal, interval, and ratio The mode (if it exists) can be used with all four levels The median can be used with data at the ordinal level of above. The mean can be used with data at the interval or ratio level (there are some exceptions at the ordinal level). Be careful with taking the average of averages! We need to know the sample size used to find each individual average
Page 88 Distribution Shapes and Averages There are ―general relationships‖ among the mean, median, and mode for different types of distributions. (a) Mound-shaped symmetrical: (b) Skewed left: the (c) Skewed right: the the mean, median, and mean is less than the mode is the smallest mode are the same or almost median and the value, the median is the the same median is less than the next largest, and the mean mode is the largest
Homework Page 89 #1, 2, 3, 6, 7, 8, 14 #17 Parts a, b, & c only #18
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