WEBVTT
00:00:00.570 --> 00:00:04.220
The table shows the marks that 10 students received in history and geography.
00:00:05.400 --> 00:00:09.450
Calculate the Pearsonβs correlation coefficient and determine the type of correlation.
00:00:10.930 --> 00:00:14.950
Letβs start by recalling the equation for the Pearsonβs correlation coefficient.
00:00:15.710 --> 00:00:51.960
So we have that the Pearsonβs correlation coefficient or π is equal to π timesed by the sum from one to π of π₯π times π¦π minus the sum from one to π of π₯π timesed by the sum from one to π of π¦π all over the square root of π lots of the sum from one to π of π₯π squared minus the sum from one to π of π₯π all squared multiplied by π timesed by the sum from one to π of π¦π squared minus the sum from one to π of π¦π all squared.
00:00:53.170 --> 00:00:54.550
Now this looks very scary.
00:00:54.750 --> 00:01:01.310
So weβll calculate each component individually and then substitute them back into the equation to find our answer.
00:01:02.250 --> 00:01:04.160
Now letβs redraw our table.
00:01:04.690 --> 00:01:13.640
But this time, weβll include columns for π₯π times π¦π, π₯π squared, and π¦π squared.
00:01:17.540 --> 00:01:18.350
Here is our table.
00:01:19.280 --> 00:01:24.060
We have labelled the history results as π₯π and the geography results as π¦π.
00:01:25.530 --> 00:01:28.900
Now letβs quickly note that π is just a number of students.
00:01:29.170 --> 00:01:31.620
And so in our case, π equals 10.
00:01:32.270 --> 00:01:34.590
Now letβs fill out some of the columns in our table.
00:01:35.180 --> 00:01:44.050
For π₯ππ¦π, we simply take the history result π₯π and multiply it by the geography result π¦π for each individual student.
00:01:44.110 --> 00:01:45.170
This is what it will look like.
00:01:46.230 --> 00:01:48.090
Next, letβs work out π₯π squared.
00:01:48.190 --> 00:01:50.900
So we simply take every π₯π value and square it.
00:01:51.980 --> 00:01:53.310
This is what we will get.
00:01:53.820 --> 00:01:58.100
Now letβs work out the final column, which we get by squaring the π¦π values.
00:01:59.140 --> 00:02:01.630
Now this is what our completed table should look like.
00:02:02.270 --> 00:02:05.850
And weβre now ready to calculate the values in the equation.
00:02:06.850 --> 00:02:15.270
So for the sum from one to π of π₯ππ¦π, we simply add together all the values in the π₯ππ¦π column in the table.
00:02:16.290 --> 00:02:17.960
So thatβs this column here.
00:02:21.730 --> 00:02:25.290
This gives us 65061.
00:02:26.310 --> 00:02:29.750
Next, weβll calculate the sum from one to π of π₯π.
00:02:30.410 --> 00:02:37.570
So we simply add together all of these values in the table, since they are in the π₯π column.
00:02:38.670 --> 00:02:40.870
This gives us 784.
00:02:41.980 --> 00:02:48.140
Now for the sum from one to π of π¦π, we simply add together all the values in the π¦π column of the table.
00:02:48.940 --> 00:02:52.390
This gives us 837.
00:02:53.430 --> 00:03:00.830
Now for the sum from one to π of π₯π squared, we simply add together the π₯π squared terms in the table.
00:03:01.030 --> 00:03:02.070
So thatβs this column here.
00:03:03.250 --> 00:03:06.610
And this gives us 62752.
00:03:08.110 --> 00:03:15.650
Now for the sum from one to π of π¦π squared, we simply add together all the values in the π¦π squared column.
00:03:15.840 --> 00:03:16.870
So thatβs this column here.
00:03:17.700 --> 00:03:22.200
And this gives us 70565.
00:03:22.990 --> 00:03:30.230
We will also need to calculate the sum from one to π of π₯π all squared and the sum from one to π of π¦π all squared.
00:03:30.540 --> 00:03:38.900
And we do this by squaring the values which weβve already calculated for the sum from one to π of π₯π and the sum from one to π of π¦π.
00:03:39.860 --> 00:03:46.740
And this gives us 614656 and 700569.
00:03:48.160 --> 00:03:54.070
So now we have calculated all the components of our Pearsonβs correlation coefficient equation.
00:03:54.130 --> 00:03:57.710
And so we can put them into the equation to find our coefficient.
00:03:58.950 --> 00:04:26.940
So substituting these values in, we get that π is equal to 10 times 65061 minus 784 times 837 all divided by the square root of 10 lots of 62752 minus 614656 times 10 lots of 70565 minus 700569.
00:04:27.870 --> 00:04:35.940
Now we just need to type this into our calculator to give us an answer of π is equal to minus 0.6924.
00:04:36.950 --> 00:04:40.020
Now we are also asked to determine the type of correlation.
00:04:41.140 --> 00:04:46.240
And since our answer is negative, therefore, this means there is an inverse correlation.