Means & Medians Chapter 4

Parameter - Fixed value about a population Typical unknown

Statistic - Value calculated from a sample

Measures of Central Tendency Median - the middle of the data; 50th percentile Observations must be in numerical order Is the middle single value if n is odd The average of the middle two values if n is even NOTE: n denotes the sample size

Measures of Central Tendency parameter Mean - the arithmetic average Use m to represent a population mean Use x to represent a sample mean statistic Formula: S is the capital Greek letter sigma – it means to sum the values that follow

Measures of Central Tendency Mode – the observation that occurs the most often Can be more than one mode If all values occur only once – there is 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 we have sample of 6 customers that buy the following number of lollipops. The median is … 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

Suppose we have sample of 6 customers that buy the following number of lollipops. 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 would happen to the median & mean if the 12 lollipops were 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 would happen to the median & mean if the 20 lollipops were 50? 5 The median is . . . 12.17 The mean is . . . What happened? 2 3 4 6 8 50

Resistant - YES NO Statistics that are not affected by outliers Is the median resistant? YES Is the mean resistant? NO

YES Look at the following data set. Find the mean. 22 23 24 25 25 26 29 30 Now find how each observation deviates from the mean. What is the sum of the deviations from the mean? Will this sum always equal zero? This is the deviation from the mean. YES

27 27 Look at the following data set. Find the mean & median. Mean = Create a histogram with the data. (use x-scale of 2) Then find the mean and median. Look at the placement of the mean and median in this symmetrical 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 Look at the following data set. Find the mean & median. Create a histogram with the data. (use x-scale of 8) Then find the mean and median. Look at the placement of the mean and median in this right skewed distribution. Use scale of 2 on graph 22 29 28 22 24 25 28 21 25 23 24 23 26 36 38 62 23

Create a histogram with the data. Then find the mean and median. Look at the following data set. Find the mean & median. Mean = Median = 54.588 58 Create a histogram with the data. Then find the mean and median. Look at the placement of the mean and median in this skewed left 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: In a symmetrical distribution, the mean and median are equal. In a skewed distribution, the mean is pulled in the direction of the skewness. In a symmetrical distribution, you should report the mean! In a skewed distribution, the median should be reported as the measure of center!

Trimmed mean: To calculate a trimmed mean: Multiply the % to trim by n Truncate that many observations from BOTH ends of the distribution (when listed in order) Calculate the mean with the shortened data set

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!