Mode, Median, Mean, Range, and Standard Deviation (1.3)

in this video we will be looking at the mode median mean range and standard deviation in the last video you saw how we can display a dataset using things like histograms stem plots and pie charts we use these diagrams to help us visually display a data set however there is another way we can describe a distribution and that way is by using numbers the mode median mean range and standard deviation give us numerical information about the distribution of a data set specifically the mode median and mean are measures of center or central tendency and the

range and standard deviation are measures of spread we will look at how we can determine the measures of center first so suppose I took a random sample of nine people and measured their heights now the mode of a dataset refers to the data value that is most frequently observed notice how the number 154 appears three times in this data set this means that the mode of this data set is equal to 154 now the median refers to the data value that is positioned in the middle of an ordered data set students often forget that to

find the median your data must be first put into order we usually order the data set from smallest to largest we can clearly see that the number 154 is in the middle of the data set because there are four data points above it and there are four data points below it so the median of this data set is equal to 154 when a data set is extremely large it might be helpful for us to use the formula n plus 1/2 this formula tells us the position of the median n refers to the total number of

data values in our sample we have a total of nine values in our sample so n is equal to nine and nine plus one is equal to ten and ten divided by two gives us five as a result the median is equal to the value in the fifth position which is equal to 154 note that we could have counted from the bottom and we would still get the same answer as long as the data set is ordered we always use the formula n plus 1/2 to find the position of a median when we have an

odd amount of data values the median will always be a parent however if n is an even number we see that we have two middle data points and if we use the formula we end up with 5.5 we see that there isn't a value in this position so what we do is we take the arithmetic average of the two values beside this position so we would have 154 plus 155 divided by 2 and we get an answer of 154 point five this value is in fact the value of the median for this data set the

last measure of central tendency we will talk about is the mean the mean is just another name for the arithmetic average the formula of the mean is as follows it is equal to the summation of all data values divided by the total number of data values if our mean comes from a sample we call it X bar so to get the mean for this sample we add up all the data values and since there is a total of ten values we will divide by ten as a result we get a mean or x-bar of 165

point 6 let's quickly compare between the median and the mean both of these are measures of center but they measure Center in a different way the median refers to the physical middle point so for this data set the median would be equal to 12 now the mean can be thought of as the balance point if you calculate the mean for this data set you would get a value of 10 if these people were of equal weights this is the position in which a seesaw would be balanced now let's talk about the measures of spread this

includes the range and standard deviation both of these values measures spread in a different way the range is simply the maximum minus the minimum so it tells us how much room a distribution takes in this data set the range is equal to the largest number which is 196 minus the smallest number which is 139 as a result the range is equal to 57 now the standard deviation is computed using this formula the formula looks a little complicated but the calculation for the standard deviation is simple we will calculate the standard deviation for the following data

set I will create a table to help me with my calculations and this table corresponds to the numerator of the formula notice how X bar is contained within the formula so we should calculate this first you should find that the mean is equal to fifteen point four the formula says we need to subtract each value from X bar so we do this on the table ten minus fifteen point four is negative five point four 12 minus fifteen point four is negative three point four 16 minus fifteen point four is zero point six and so on

the next step is to square what we have just calculated negative five point four squared is twenty nine point sixteen negative three point four squared is eleven point fifty six and so on the next step is to find the sum of what we have just calculated you should find that this is equal to seventy five point to remember that we use this table to calculate the numerator of the formula so we can now replace it with seventy five point two from here the formula should be pretty straightforward and refers to the total number of data

values and there are five data values in this data set so n is equal to five we can simplify this and we end up with a standard deviation that is equal to four point three three six now what does the standard deviation even tell us the standard deviation tells us how close the values in a data set are to the mean for example a small standard deviation indicates a small amount of variability for a given data set in other words there will be a lot of values that are closer to the mean which makes the

distribution less spread out in contrast a high standard deviation indicates a high amount of variability for a given data set in other words there will be a lot more values that are farther from the mean which makes the distribution more spread out the last thing we will talk about is variance variance is closely related to the standard deviation the only difference between these two formulas is that the standard deviation involves taking the square root of the calculations and for the variance we don't take the square root also notice how for the standard deviation we denote

it as s and for the variance we denote it as s squared both of these can be referred to as the sample variance and the sample standard deviation