Hi, I'm Dr. Todd Daniel, and you're watching "Statistics for the Flipped
Classroom." Last time, I showed you how to find the center of the data. The next
thing we want to know is how close together or far apart our scores are,
because when our scores are close together, our measure of central tendency
is more representative of the rest of the data. So we're going to start simply,
by learning about the range, then the inner quartile range, then the
five-number summary. Then we're going to get mathematical, and I'll teach you how
to compute the sum of squares, the variance, and the standard deviation so
let's get started.
We have two important questions about our data set. The first question is what
single number best represents our data? The second question is whether the
scores are packed together or spread out? The answer to the second question about
scores being close together or spread out is going to be some measure of
variability. Variability indicates how spread out the scores are, how much the
scores differ from each other. When there are large differences among the scores,
the data are said to contain a lot of variability. The opposite of variability
is consistency. Consistency is the term that we use to describe variability in
everyday life. If people, events, and experiences are very similar, we call
them consistent, and as human beings we like consistency, because we like to know
what to expect. Consistency minimizes uncertainty, so imagine that you are
driving cross-country and you decide to stop and eat a hamburger at every
McDonald's that you pass. Do you expect the variability of the hamburgers from
one restaurant to the next to be high or low? Will the consistency be high or low?
The variability will be low. Consistency will be high. Every burger will taste
like every other burger, and that is why we like low variability or high
consistency. We like consistency in people, events, and experiences, and data.
What do you call a person with high consistency? Stable, steady, predictable.
What do you call a person with high variability? Unpredictable, inconsistent,
bipolar. Do you like having friends who when they tell you they will be at a
certain place at a certain time you know you can count on them? They are
consistent. They are low drama and dependable, and that is what we like in
our data too. You see, the greater the variability, the less accurately a
measure of central tendency summarizes the distribution. Predictions are less
precise. Measurement error is larger. The mean is not as useful. But when the data
are consistent with the mean, those data are predictable, stable, and
representative. Low variability equals high predictability and high consistency.
Later on, when we talk about t-tests I will remind you of this when we test for
homogeneity of variance. When we compare two groups, it is important that one
group not be high variance and the other group low variance. The nature of the
variance needs to be similar for us to be able to analyze properly. It is
important to remember that variability and central tendency measure two
different things, and they are independent of each other. The mean tells
us where the center of the data is, but distributions with the same mean may
have different variability. So for example, what is the mean of this first
data set? The mean is 6, but look at the highest and lowest numbers. There's a lot
of variability here. Now this next data set also has a mean of 6, but the numbers
are closer together. This last data set has no variability. The mean is still six,
but every score in the data set is highly representative of the mean. The
mean is a perfect predictor. If I choose an X score at random and ask you to
predict what it will be, you will get it exactly right every time. The measurement
error in the first data set is much higher.
Using the mean to predict a highly variable data set is much less useful,
because the score is less representative but with low variability, we have high
predictability.