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##### Central Tendency

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**Central Tendency**Statistics 2126**Introduction**• As useful as like histograms and such are, it would be nice to describe data in terms of Central Tendency • A single number to describe a sample • BTW, the Sample is a subset of the population • We are almost always dealing with samples**Back when I was in first year…**• 77 80 83 70 90 • Would be nice to describe how I did in first year with a number • Well the one we are all pretty used to is the mean or arithmetic average • The sum of all of the data points, divided by the number of data points**The Mean**• Sort of a balancing point in the data • Simply adding up the numbers and dividing by the number of observations (n) • X bar is for the sample • We might want to consider my first year marks as a population**For a population**• The formula does not change, but the symbol does • We use statistics for samples • We use parameters for populations • • The formula is the same really**The mean is not mean**• In the population, the mean does not change • The sample, yeah it changes, sample to sample • Parameters do NOT CHANGE**However, the lecture is getting meaner**• If you sample from a population you will get different values for x bar each time • We don’t care about samples in the long run, we care about populations • Calculating is pretty hard, umm it takes forever • Used sometimes, elections, the census**Samples vs. populations**• A good sample will give you a killer estimate of the population • The census could be done via sampling actually • This is because x bar is an unbiased estimator of • It overestimates as often as it underestimates**Weighted averages sometimes**• Some assignments worth more than others for example • There are other measures of central tendency though**The median**• No need for a formula here • 50th percentile • Midpoint • Half below, half above**The mode**• The most common observation • Virtually useless • Example 25 25 37 42 25 • The mode is 25 • Tough eh…**If….**• If the median = mean = mode we have a unimodal, symmetrical distribution • Say IQ in the population, all measures of central tendency = 100**Normal distribution**• You don’t have to get a normal distribution when you have a unimodal, symmetrical distribution • It is probably the most common one though**Why?**• Why do we need all of these measures of central tendency? • They all have different properties • The mode is useless… • So let’s move on**Median vs. the Mean**• Say you have five numbers • 1 2 3 4 5 • The mean is 3, as is the median • (BTW, the mode is umm well there are 5 of them) • Add another value • 750**Mean vs median in a final all out battle to the death**• Now the mean is 127.5 • So adding an extreme value really affects the mean • Median is now umm let’s see • 1 2 3 4 5 750 • 3.5 • cool**Median for the win**• So sometimes it is good • Think about say union negotiations • Both sides can talk about average salary • Both are right! • In this case the median is more useful**So the median is useful**• Especially when there are outliers • However you want to leave them in • When you want to take all of the scores into account though you have to use the mean really • All of our techniques are about means • The median is, pretty much, a dead end statistically**Running out of pithy titles**• The mean is most useful for symmetrical distributions • Most distributions we deal with will be like this • Most are pretty much symmetrical, more or less