AP Statistics Monday, 14 September 2015 OBJECTIVE TSW identify measures of central tendancy. Get a TI-83/TI-84 (mine or yours). TESTS are graded. Next QUIZ: Means, Medians, Boxplots – Friday, 18 September 2015 Next TEST: Displaying Data – Wednesday, 23 September 2015

Means & Medians

P ARAMETER Fixed value about a population Typically unknown

S TATISTIC Value calculated from a sample

M EASURES OF C ENTRAL T ENDENCY Median - the middle of the data; 50 th percentile The observations must be in numerical order. The median is the middle value if n is odd. The median is the mean of the middle two values if n is even. NOTE: n denotes the sample size

Mean - the arithmetic average Use  (mu) to represent a population mean Use x (x-bar) to represent a sample mean  Formula:  is the capital Greek letter sigma – it means to sum the values that follow parameter statistic M EASURES OF C ENTRAL T ENDENCY

Mode – the observation that occurs the most often There CAN be more than one mode. If all values occur only once – there is no mode. It is not used as often as mean & median. M EASURES OF C ENTRAL T ENDENCY

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.

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?

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

AP Statistics Tuesday, 15 September 2015 OBJECTIVE TSW identify measures of central tendancy. Get a TI-83/TI-84 (mine or yours). CHANGE: Assignment due Tomorrow – WS Means and Medians Next QUIZ: Means, Medians, Variability – Friday, 18 September 2015 Next TEST: Displaying Data – Wednesday, 23 September 2015 T-Shirts???

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 an 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 an x-scale of 8). Then find the mean and median Look at the placement of the mean and median in this right-skewed (positively 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 left- skewed (negatively skewed) distribution.

R ECAP 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!

T RIMMED 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 n n = 10, and 10%(10) = 1 So remove one observation from each side!

Assignment WS Mean and Median Due Wednesday, 16 September Next QUIZ: Means, Medians, Variability – Friday, 18 September 2015 Next TEST: Displaying Data – Wednesday, 23 September 2015