Means - Means& Medians Chapter 5 Parameter ­ Parameter...

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Unformatted text preview: Means & Medians Chapter 5 Parameter ­ Parameter ­ ►Fixed value about a population ►Typical unknown Statistic ­ Statistic ­ ►Value calculated from a sample Measures of Central Tendency 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 Measures of Central Tendency parameter ►Mean ­ the arithmetic average Use µ to represent a population statistic mean Use x to represent a sample mean Formula: ∑x x= n Σ is the capital Greek letter sigma – it means to sum the values that follow Measures of Central Tendency 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 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 Suppose we have sample of 6 customers that buy the following number of lollipops. The median is … The median is The numbers are in order 5 lollipops! & n is even – so find the middle two observations. Now, average these two values. 5 2 3 4 6 8 12 Suppose we have sample of 6 customers that buy 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. x = 5.833 2 + 3 + 4 + 6 + 8 + 12 6 2 3 4 6 8 12 Using the calculator . . . Using the calculator . . . What would happen to the median & mean if the What would happen to the median & mean if the 12 lollipops were 20? The median is . . . The mean is . . . 5 7.17 2 + 3 + 4 + 6 + 8 + 20 6 What happened? 2 3 4 6 8 20 What would happen to the median & mean if the What would happen to the median & mean if the 20 lollipops were 50? The median is . . . The mean is . . . 5 12.17 2 + 3 + 4 + 6 + 8 + 50 6 What happened? 2 3 4 6 8 50 Resistant ­ Resistant ­ ►Statistics that are not affected by outliers ►Is the median resistant? ►Is the mean resistant? YES NO Look at the following data set. Find the mean. 22 23 24 25 25 26 29 30 x = 25.5 Will this sum always Now find how each observation equal zero? deviates from the mean. This is the deviation from the mean. YES What is the sum of the deviations from the mean? ∑ (x − x ) = 0 Look at the following data set. Find the Look at the following data set. Find the mean & median. Mean = 27 Median = 27 21 27 Create a histogram with the data. (use x­scale of 2) Then Look at the placement of find the mean and median. the mean and median in this symmetrical distribution. 25 23 23 24 25 27 27 28 30 30 26 26 26 27 30 31 32 32 Look at the following data set. Find the Look at the following data set. Find the mean & median. Mean = 28.176 Median = 25 Create a histogram with the data. (use x­scale of 8) Then Look at the placement of find the mean and median. the mean and median in this right skewed 22 29 distribution. 24 28 22 25 28 21 23 62 23 24 23 26 36 38 25 Look at the following data set. Find the Look at the following data set. Find the mean & median. Mean = 54.588 Median = 58 Create a histogram with the data. Then find the mean and Look at the placement of median. the mean and median in this skewed left distribution. 53 21 46 54 47 60 55 55 56 63 64 58 58 58 58 62 60 Recap: 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: 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. Find a 10% trimmed mean with the following data. 12 14 19 20 22 24 25 26 26 10%(10) = 1 So remove one observation from each side! 14 + 19 + 20 + 22 + 24 + 25 + 26 + 26 = 22 8 35 ...
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