Means & Medians Chapter 5

Parameter - ► Fixed value about a population ► Typical unknown

Statistic - ► Value calculated from a sample

Measures of Central Tendency ► Median - the middle of the data; 50 th 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 ► Mean - the arithmetic average  Use μ to represent a population mean  Use x to represent a sample mean  Formula: Σ is the capital Greek letter sigma – it means to sum the values that follow parameter statistic

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!

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

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.

Using the calculator...

What would happen to the median & mean if the 12 lollipops were 20? The median is... 5 The mean is What happened?

What would happen to the median & mean if the 20 lollipops were 50? The median is... 5 The mean is What happened?

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

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

Look at the following data set. Find the mean & median. Mean = Median = Create a histogram with the data. (use x-scale of 2) Then find the mean and median. 27 Look at the placement of the mean and median in this symmetrical distribution.

Look at the following data set. Find the mean & median. 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.

Look at the following data set. Find the mean & median. Mean = Median = 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.

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

Find a 10% trimmed mean with the following data %(10) = 1 So remove one observation from each side!