Introduction to Descriptive Statistics

right now you see everything that you need to know about the data but it's not very satisfying because it's all disorganized descriptive statistics allow you to organize the data so we're going to make a frequency distribution a frequency histogram and calculate some measures of central tendency and dispersion for this set of data we begin by making a frequency distribution for a frequency distribution you make a table the table lists every value of the data that you have and it's easier if you put it in order so if we look at this data we see oh we've got a 1 and there's a 2 3 4 5 so those are the possible values for each of these data points then we count the frequency of each so there's one one and there's two twos and there's one two three four threes and if we look we see two fours and the final one is one five this is the frequency distribution in a table form we can make it into a graph by putting the x and y coordinates the score is X 1 2 3 4 5 and the y axis is equal to the frequency over here so we had 1 2 3 4 let's make five levels on our frequency there was one one so we make a bar that's one unit high one frequency there were two twos so we make a bar that goes up to two there were four three two fours and one five and this is a frequency histogram that organizes the data and tells us much more about it but we want to know more about it what is the center of this distribution and how dispersed is it how spread out is the center of that distribution we calculate measures of central tendency there are three different ways that we can do this the easiest is the mode the mode is the most frequent we can see from either the frequency table or from the frequency histogram that three is the most frequent score that's the mode the median is the middle score and so we have to have the same number above and below well how many scores are there there are ten scores so the median is going to be between five and six well there's one that's one and less three of them if you look and say okay there were 2 2 2 1 1 3 & 4 so we know the median is going to be between the 2nd and the 3rd 3 but what's the difference between 3 & 3 oh it's 3 the median is the middle score there is an equal number above and below it approximate the mean is the average and we calculated it by taking the sum of all the scores divided by the number of scores so we have to add up each of the scores we can add them here we can add them in the frequency table if we remember that we have 1 1 that's 1 & 2 2 so 1 plus 2 is 3 plus 2 makes another 5 we add these all up and we're going to find that we have a sum of 30 and there are 10 scores so the mean is equal to 3 these are three measures of central tendency in addition to knowing the central tendency of the data what's the middle we also like to know how spread out the data are and you can look at the frequency histogram and get an idea but we'd like a number for it so we calculate what is called dispersion in order kept to calculate the dispersion we need to know how far each score is away from the mean we call that the deviation score and that's an italicized a deviation score is the raw score minus the mean which we're calling x-bar if we calculate a deviation score for each score in the distribution we could have a bunch of those deviation scores but if we calculate their average to see how spread out it is on average it will always equal zero because if you look at this 1 minus 3 is going to give us 2 but 5 minus 3 is also going to give us 2 and this is a minus 2 and a plus 2 so in order to solve that problem statisticians that let's just square it because if you take a minus 2 times a minus 2 you get 4 a plus 4 so the deviation scores squared divided by not the entire number of n minus 1 this is called a correction for sampling error is what is equal to the variance in order to calculate the variance we have to add up each of these deviation scores squared and then divide by the number minus one this is a correction for sampling error so if we look at our frequencies we can see that we have 1 1 so 1 minus 3 is equal to 2 2 squared is equal to 4 so that's the deviation score squared for that there are two 2's the deviation from two minus one is a minus one minus one squared is one and there are two of them so we put them in twice now we get to the most frequent score which is also the mean 3 3 minus 3 is 0 0 squared is 0 but there are 4 of them so it's 0 plus 0 plus 0 plus 0 now we're up to 4 4 minus 3 is 1 1 squared is 1 and there are two fours so we put it in twice and finally we get to the last score 5 5 minus 3 is 2 2 squared is 4 and so we have 1 4 1 squared is equal to 1 1 all of this is divided by n minus 1 there are 10 scores at the end minus 1 so the variance is equal to this sum we add them up four plus one plus one is six plus these four zeroes is 6 plus one plus one plus four the total is equal to 12 divided by 10 minus 1 which is 9 which is equal to one point three three now we are not happy with this number because this is in a squared units and we don't think about things as squared so statisticians have come up with a way to get rid of that and they call it the standard deviation the standard deviation is equal to the square root of the variance so in order to calculate the standard deviation for this particular set of data we simply have to take one point three three and calculate the square root now I can't do that in my head so I used a calculator earlier and I got one point one five so now we have taken some disorganized data organize it into a frequency table made a frequency histogram calculated three measures of central tendency and we have a measure of dispersion in terms of the same units that we started out with in the standard deviation

Channel: Sandra Webster

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Keywords: FlipShare, educational, tutorial, statistics, histogram, mean, median, mode, variance, dispersion

Id: MhDH9jsyzBA

Length: 8min 59sec (539 seconds)

Published: Wed Mar 16 2011