Means & Medians Chapter 4

Parameter A value describing a population Typically unknown

Statistic A value calculated from sample data

Measures of Central Tendency Median: middle of the data (50th percentile) Put data in numerical order n is odd  median = middle number n is even  median = average of middle two numbers (n = sample size)

Measures of Central Tendency parameter Mean: arithmetic average m (mu) - population mean x (x-bar) - sample mean statistic S – capital Greek letter sigma – sum up the values that follow Formula:

Measures of Central Tendency Mode – observation that occurs most Can have more than one mode Can have no mode Not used as often as mean & median

Suppose we are interested in the number of lollipops that are bought at a certain store. A sample of 5 customers buys the following number of lollipops. Find the median. The numbers are in order & n is odd – so find the middle observation. The median is 4 lollipops! 2 3 4 8 12

Suppose a sample of 6 customers buy the following numbers of lollipops Suppose a sample of 6 customers buy the following numbers of lollipops. Find the median. The numbers are in order & n is even – so find the middle two observations. The median is 5 lollipops! Now, average these two values. 5 2 3 4 6 8 12

Now find the mean. To find the mean number of lollipops, add the observations and divide by n. 2 3 4 6 8 12

Using the calculator . . .

2 3 4 6 8 20 5 7.17 The median is . . . The mean is . . . What if the 12 lollipops became 20? 5 The median is . . . 7.17 The mean is . . . What happened? 2 3 4 6 8 20

2 3 4 6 8 50 5 12.17 The median is . . . The mean is . . . What if the 12 lollipops became 50? 5 The median is . . . 12.17 The mean is . . . What happened? 2 3 4 6 8 50

Resistant YES NO Resistant statistics are not affected by outliers Is the median resistant? YES Is the mean resistant? NO

Sum of Deviations YES Find the mean of the following data. 22 23 24 25 25 26 29 30 Will this sum always equal zero? Find the sum of the deviations of each value from the mean. Deviation from the mean YES Mean = balance point

27 27 Find the mean and median of the data set. Mean = Median = Make a histogram with an x-scale of 2 What shape is this distribution? Use scale of 2 on graph 21 23 23 24 25 25 26 26 26 27 27 27 27 28 30 30 30 31 32 32

28.176 25 Find the mean and median of the data set. Mean = Median = Make a histogram with an x-scale of 8 What shape is this distribution? Use scale of 2 on graph 22 29 28 22 24 25 28 21 25 23 24 23 26 36 38 62 23

What shape is this distribution? Find the mean and median of the data set. Mean = Median = 54.588 58 Make a histogram What shape is this distribution? Use scale of 2 on graph 21 46 54 47 53 60 55 55 60 56 58 58 58 58 62 63 64

Recap: Skewed: mean is pulled toward the skewness Symmetrical: mean = median Report the mean as the center Skewed: mean is pulled toward the skewness Report the median as the center

Trimmed mean: List data in order Multiply the % to trim by n Cut off that many observations from BOTH ends of the data Calculate the new mean

So remove one observation from each side Find a 10% trimmed mean with the following data. 12 14 19 20 22 24 25 26 26 35 10%(10) = 1 So remove one observation from each side