PL9fwy3NUQKwaIc7KWc8buW344941GStQL/EuXnSmrCFuE_raw.srt

1
00:00:06,480 --> 00:00:12,100
Last time, I mean Tuesday, we discussed box plot

2
00:00:12,100 --> 00:00:19,540
and we introduced how can we use box plot to

3
00:00:19,540 --> 00:00:24,160
determine if any point is suspected to be an

4
00:00:24,160 --> 00:00:28,280
outlier by using the lower limit and upper limit.

5
00:00:29,460 --> 00:00:32,980
And we mentioned last time that if any point is

6
00:00:32,980 --> 00:00:38,580
below the lower limit or is above the upper limit,

7
00:00:39,200 --> 00:00:43,000
that point is considered to be an outlier. So

8
00:00:43,000 --> 00:00:46,940
that's one of the usage of the backsplat. I mean,

9
00:00:47,000 --> 00:00:51,360
for this specific example, we mentioned last time

10
00:00:51,360 --> 00:00:56,910
27 is an outlier. And also here you can tell also

11
00:00:56,910 --> 00:01:02,770
the data are right skewed because the right tail

12
00:01:02,770 --> 00:01:06,090
exactly is much longer than the left tail. I mean

13
00:01:06,090 --> 00:01:10,450
the distance between or from the median and the

14
00:01:10,450 --> 00:01:14,150
maximum value is bigger or larger than the

15
00:01:14,150 --> 00:01:16,950
distance from the median to the smallest value.

16
00:01:17,450 --> 00:01:20,870
That means the data is not symmetric, it's quite

17
00:01:20,870 --> 00:01:24,050
skewed to the right. In this case, you cannot use

18
00:01:24,050 --> 00:01:29,370
the mean or the range as a measure of spread and

19
00:01:29,370 --> 00:01:31,730
median and, I'm sorry, mean as a measure of

20
00:01:31,730 --> 00:01:36,130
tendency. Because these measures are affected by

21
00:01:36,130 --> 00:01:39,450
outcomes. In this case, you have to use the median

22
00:01:39,450 --> 00:01:43,690
instead of the mean and IQR instead of the range

23
00:01:43,690 --> 00:01:48,090
because IQR is the mid-spread of the data because

24
00:01:48,090 --> 00:01:52,790
we just take the range from Q3 to Q1. That means

25
00:01:52,790 --> 00:01:57,450
we ignore The data below Q1 and data after Q3.

26
00:01:57,970 --> 00:02:01,370
That means IQR is not affected by outlier and it's

27
00:02:01,370 --> 00:02:04,610
better to use it instead of R, of the range.

28
00:02:07,470 --> 00:02:10,950
If the data has an outlier, it's better just to

29
00:02:10,950 --> 00:02:13,990
make a star or circle for the box plot because

30
00:02:13,990 --> 00:02:17,250
this one mentioned that that point is an outlier.

31
00:02:18,390 --> 00:02:21,390
Sometimes outlier is maximum value or the largest

32
00:02:21,390 --> 00:02:25,000
value you have. sometimes maybe the minimum value.

33
00:02:25,520 --> 00:02:28,480
So it depends on the data. For this example, 27,

34
00:02:28,720 --> 00:02:33,360
which was the maximum, is an outlier. But zero is

35
00:02:33,360 --> 00:02:36,520
not outlier in this case, because zero is above

36
00:02:36,520 --> 00:02:41,500
the lower limit. Let's move to the next topic,

37
00:02:42,140 --> 00:02:48,060
which talks about covariance and correlation.

38
00:02:48,960 --> 00:02:51,740
Later, we'll talk in more details about

39
00:02:53,020 --> 00:02:56,060
Correlation and regression, that's when maybe

40
00:02:56,060 --> 00:03:02,840
chapter 11 or 12. But here we just show how can we

41
00:03:02,840 --> 00:03:05,420
compute the covariance of the correlation

42
00:03:05,420 --> 00:03:10,220
coefficient and what's the meaning of that value

43
00:03:10,220 --> 00:03:15,840
we have. The covariance means it measures the

44
00:03:15,840 --> 00:03:21,090
strength of the linear relationship between two

45
00:03:21,090 --> 00:03:25,410
numerical variables. That means if the data set is

46
00:03:25,410 --> 00:03:29,770
numeric, I mean if both variables are numeric, in

47
00:03:29,770 --> 00:03:33,050
this case we can use the covariance to measure the

48
00:03:33,050 --> 00:03:38,390
strength of the linear association or relationship

49
00:03:38,390 --> 00:03:42,310
between two numerical variables. Now the formula

50
00:03:42,310 --> 00:03:45,330
is used to compute the covariance given by this

51
00:03:45,330 --> 00:03:52,540
one. It's summation of the product of xi minus x

52
00:03:52,540 --> 00:03:56,380
bar, yi minus y bar, divided by n minus 1.

53
00:03:59,660 --> 00:04:03,120
So we need first to compute the means of x and y,

54
00:04:03,620 --> 00:04:07,680
then find x minus x bar times y minus y bar, then

55
00:04:07,680 --> 00:04:11,160
sum all of these values, then divide by n minus 1.

56
00:04:12,870 --> 00:04:17,770
The covariance only concerned with the strength of

57
00:04:17,770 --> 00:04:23,370
the relationship. By using the sign of the

58
00:04:23,370 --> 00:04:27,010
covariance, you can tell if there exists positive

59
00:04:27,010 --> 00:04:31,070
or negative relationship between the two

60
00:04:31,070 --> 00:04:33,710
variables. For example, if the covariance between

61
00:04:33,710 --> 00:04:42,760
x and y is positive, that means x and y move In

62
00:04:42,760 --> 00:04:48,080
the same direction. It means that if X goes up, Y

63
00:04:48,080 --> 00:04:52,260
will go in the same position. If X goes down, also

64
00:04:52,260 --> 00:04:55,660
Y goes down. For example, suppose we are

65
00:04:55,660 --> 00:04:57,920
interested in the relationship between consumption

66
00:04:57,920 --> 00:05:02,440
and income. We know that if income increases, if

67
00:05:02,440 --> 00:05:07,160
income goes up, if your salary goes up, that means

68
00:05:07,160 --> 00:05:13,510
consumption also will go up. So that means they go

69
00:05:13,510 --> 00:05:18,650
in the same or move in the same position. So for

70
00:05:18,650 --> 00:05:20,690
sure, the covariance between X and Y should be

71
00:05:20,690 --> 00:05:25,550
positive. On the other hand, if the covariance

72
00:05:25,550 --> 00:05:31,110
between X and Y is negative, that means X goes up.

73
00:05:32,930 --> 00:05:36,370
Y will go to the same, to the opposite direction.

74
00:05:36,590 --> 00:05:40,090
I mean they move to opposite direction. That means

75
00:05:40,090 --> 00:05:42,230
there exists negative relationship between X and

76
00:05:42,230 --> 00:05:47,630
Y. For example, you score in statistics a number

77
00:05:47,630 --> 00:05:55,220
of missing classes. If you miss more classes, it

78
00:05:55,220 --> 00:05:59,860
means your score will go down so as x increases y

79
00:05:59,860 --> 00:06:04,820
will go down so there is positive relationship or

80
00:06:04,820 --> 00:06:08,720
negative relationship between x and y i mean x

81
00:06:08,720 --> 00:06:12,020
goes up the other go in the same direction

82
00:06:12,020 --> 00:06:16,500
sometimes

83
00:06:16,500 --> 00:06:21,800
there is exist no relationship between x and y In

84
00:06:21,800 --> 00:06:24,780
that case, covariance between x and y equals zero.

85
00:06:24,880 --> 00:06:31,320
For example, your score in statistics and your

86
00:06:31,320 --> 00:06:31,700
weight.

87
00:06:34,540 --> 00:06:36,760
It makes sense that there is no relationship

88
00:06:36,760 --> 00:06:42,680
between your weight and your score. In this case,

89
00:06:43,580 --> 00:06:46,760
we are saying x and y are independent. I mean,

90
00:06:46,840 --> 00:06:50,790
they are uncorrelated. Because as one variable

91
00:06:50,790 --> 00:06:56,010
increases, the other maybe go up or go down. Or

92
00:06:56,010 --> 00:06:59,690
maybe constant. So that means there exists no

93
00:06:59,690 --> 00:07:02,390
relationship between the two variables. In that

94
00:07:02,390 --> 00:07:05,950
case, the covariance between x and y equals zero.

95
00:07:06,450 --> 00:07:09,210
Now, by using the covariance, you can determine

96
00:07:09,210 --> 00:07:12,710
the direction of the relationship. I mean, you can

97
00:07:12,710 --> 00:07:14,850
just figure out if the relation is positive or

98
00:07:14,850 --> 00:07:18,980
negative. But you cannot tell exactly the strength

99
00:07:18,980 --> 00:07:22,100
of the relationship. I mean you cannot tell if

100
00:07:22,100 --> 00:07:27,640
they exist. strong moderate or weak relationship

101
00:07:27,640 --> 00:07:31,040
just you can tell there exists positive or

102
00:07:31,040 --> 00:07:33,520
negative or maybe the relationship does not exist

103
00:07:33,520 --> 00:07:36,580
but you cannot tell the exact strength of the

104
00:07:36,580 --> 00:07:40,120
relationship by using the value of the covariance

105
00:07:40,120 --> 00:07:43,060
I mean the size of the covariance does not tell

106
00:07:43,060 --> 00:07:48,520
anything about the strength so generally speaking

107
00:07:48,520 --> 00:07:53,150
covariance between x and y measures the strength

108
00:07:53,150 --> 00:07:58,590
of two numerical variables, and you only tell if

109
00:07:58,590 --> 00:08:01,270
there exists positive or negative relationship,

110
00:08:01,770 --> 00:08:04,510
but you cannot tell anything about the strength of

111
00:08:04,510 --> 00:08:06,910
the relationship. Any questions?

112
00:08:09,610 --> 00:08:15,210
So let me ask you just to summarize what I said so

113
00:08:15,210 --> 00:08:21,100
far. Just give me the summary or conclusion. of

114
00:08:21,100 --> 00:08:24,670
the covariance. The value of the covariance

115
00:08:24,670 --> 00:08:26,810
determine if the relationship between the

116
00:08:26,810 --> 00:08:29,410
variables are positive or negative or there is no

117
00:08:29,410 --> 00:08:31,970
relationship that when the covariance is more than

118
00:08:31,970 --> 00:08:34,170
zero, the meaning that it's positive, the

119
00:08:34,170 --> 00:08:36,930
relationship is positive and one variable go up,

120
00:08:37,290 --> 00:08:39,590
another go up and vice versa. And when the

121
00:08:39,590 --> 00:08:41,810
covariance is less than zero, there is negative

122
00:08:41,810 --> 00:08:44,250
relationship and the meaning that when one

123
00:08:44,250 --> 00:08:47,490
variable go up, the other goes down and vice versa

124
00:08:47,490 --> 00:08:50,550
and when the covariance equals zero, there is no

125
00:08:50,550 --> 00:08:53,350
relationship between the variables. And what's

126
00:08:53,350 --> 00:08:54,930
about the strength?

127
00:08:57,950 --> 00:09:03,450
So just tell the direction, not the strength of

128
00:09:03,450 --> 00:09:08,610
the relationship. Now, in order to determine both

129
00:09:08,610 --> 00:09:12,110
the direction and the strength, we can use the

130
00:09:12,110 --> 00:09:17,580
coefficient of correlation. The coefficient of

131
00:09:17,580 --> 00:09:20,320
correlation measures the relative strength of the

132
00:09:20,320 --> 00:09:22,780
linear relationship between two numerical

133
00:09:22,780 --> 00:09:27,940
variables. The simplest formula that can be used

134
00:09:27,940 --> 00:09:31,220
to compute the correlation coefficient is given by

135
00:09:31,220 --> 00:09:34,440
this one. Maybe this is the easiest formula you

136
00:09:34,440 --> 00:09:38,060
can use. I mean, it's shortcut formula for

137
00:09:38,060 --> 00:09:40,860
computation. There are many other formulas to

138
00:09:40,860 --> 00:09:44,490
compute the correlation. This one is the easiest

139
00:09:44,490 --> 00:09:52,570
one. R is just sum of xy minus n, n is the sample

140
00:09:52,570 --> 00:09:57,570
size, times x bar is the sample mean, y bar is the

141
00:09:57,570 --> 00:10:01,090
sample mean for y, because here we have two

142
00:10:01,090 --> 00:10:06,250
variables, divided by square root, don't forget

143
00:10:06,250 --> 00:10:11,490
the square root, of two quantities. One conserved

144
00:10:11,490 --> 00:10:15,710
for x and the other for y. The first one, sum of x

145
00:10:15,710 --> 00:10:18,850
squared minus nx bar squared. The other one is

146
00:10:18,850 --> 00:10:21,830
similar just for the other variables, sum y

147
00:10:21,830 --> 00:10:26,090
squared minus ny bar squared. So in order to

148
00:10:26,090 --> 00:10:28,650
determine the value of R, we need,

149
00:10:32,170 --> 00:10:35,890
suppose for example, we have x and y, theta x and

150
00:10:35,890 --> 00:10:36,110
y.

151
00:10:40,350 --> 00:10:44,730
x is called explanatory

152
00:10:44,730 --> 00:10:54,390
variable and

153
00:10:54,390 --> 00:11:04,590
y is called response variable

154
00:11:04,590 --> 00:11:07,970
sometimes x is called independent

155
00:11:21,760 --> 00:11:25,320
For example, suppose we are talking about

156
00:11:25,320 --> 00:11:32,280
consumption and

157
00:11:32,280 --> 00:11:36,700
input. And we are interested in the relationship

158
00:11:36,700 --> 00:11:41,360
between these two variables. Now, except for the

159
00:11:41,360 --> 00:11:44,900
variable or the independent, this one affects the

160
00:11:44,900 --> 00:11:49,840
other variable. As we mentioned, as your income

161
00:11:49,840 --> 00:11:53,800
increases, your consumption will go in the same

162
00:11:53,800 --> 00:11:59,580
direction, increases also. Income causes Y, or

163
00:11:59,580 --> 00:12:04,340
income affects Y. In this case, income is your X.

164
00:12:06,180 --> 00:12:07,780
Most of the time we use

165
00:12:10,790 --> 00:12:15,590
And Y for independent. So in this case, the

166
00:12:15,590 --> 00:12:19,370
response variable or your outcome or the dependent

167
00:12:19,370 --> 00:12:23,110
variable is your consumption. So Y is consumption,

168
00:12:23,530 --> 00:12:29,150
X is income. So now in order to determine the

169
00:12:29,150 --> 00:12:32,950
correlation coefficient, we have the data of X and

170
00:12:32,950 --> 00:12:33,210
Y.

171
00:12:36,350 --> 00:12:39,190
The values of X, I mean the number of pairs of X

172
00:12:39,190 --> 00:12:41,990
should be equal to the number of pairs of Y. So if

173
00:12:41,990 --> 00:12:44,930
we have ten observations for X, we should have the

174
00:12:44,930 --> 00:12:50,010
same number of observations for Y. It's pairs. X1,

175
00:12:50,090 --> 00:12:54,750
Y1, X2, Y2, and so on. Now, the formula to compute

176
00:12:54,750 --> 00:13:04,170
R, the shortcut formula is sum of XY minus N times

177
00:13:04,970 --> 00:13:09,630
x bar, y bar, divided by the square root of two

178
00:13:09,630 --> 00:13:12,770
quantities. The first one, sum of x squared minus

179
00:13:12,770 --> 00:13:17,270
n x bar. The other one, sum of y squared minus ny

180
00:13:17,270 --> 00:13:21,710
y squared. So the first thing we have to do is to

181
00:13:21,710 --> 00:13:24,210
find the mean for each x and y.

182
00:13:28,230 --> 00:13:37,210
So first step, compute x bar and y bar. Next, if

183
00:13:37,210 --> 00:13:41,690
you look here, we have x and y, x times y. So we

184
00:13:41,690 --> 00:13:48,870
need to compute the product of x times y. So just

185
00:13:48,870 --> 00:13:53,870
for example, suppose x is 10, y is 5. So x times y

186
00:13:53,870 --> 00:13:54,970
is 50.

187
00:13:57,810 --> 00:13:59,950
In addition to that, you have to compute

188
00:14:06,790 --> 00:14:12,470
100 x squared and y squared. It's like 125.

189
00:14:14,810 --> 00:14:18,870
Do the same calculations for the rest of the data

190
00:14:18,870 --> 00:14:22,290
you have. We have other data here, so we have to

191
00:14:22,290 --> 00:14:25,410
compute the same for the others.

192
00:14:28,470 --> 00:14:33,250
Then finally, just add xy, x squared, y squared.

193
00:14:35,910 --> 00:14:40,830
The values you have here in this formula, in order

194
00:14:40,830 --> 00:14:44,830
to compute the coefficient.

195
00:14:54,250 --> 00:15:00,070
Now, this value ranges between minus one and plus

196
00:15:00,070 --> 00:15:00,370
one.

197
00:15:06,520 --> 00:15:10,800
So it's between minus one and plus one. That means

198
00:15:10,800 --> 00:15:15,840
it's never smaller

199
00:15:15,840 --> 00:15:20,200
than minus one or greater than one. It's between

200
00:15:20,200 --> 00:15:21,480
minus one and plus one.

201
00:15:24,360 --> 00:15:28,300
Make sense? I mean if your value is suppose you

202
00:15:28,300 --> 00:15:34,520
did mistake for any of these computations and R

203
00:15:34,520 --> 00:15:41,710
might be 1.15, 115. That means there is an error.

204
00:15:42,270 --> 00:15:45,870
Or for example, if R is negative 1.5, that means

205
00:15:45,870 --> 00:15:49,610
there is a mistake. So you have to find or figure

206
00:15:49,610 --> 00:15:55,670
out what is that mistake. So that's simple

207
00:15:55,670 --> 00:15:59,090
calculations. Usually in the exam, we will give

208
00:15:59,090 --> 00:16:01,350
the formula for the correlation coefficient, as we

209
00:16:01,350 --> 00:16:03,590
mentioned before. In addition to that, we will

210
00:16:03,590 --> 00:16:04,330
give the summation.

211
00:16:07,780 --> 00:16:12,720
The sum of xy is given, sum x squared and sum y

212
00:16:12,720 --> 00:16:18,320
squared. Also sum of x and sum of y, in order to

213
00:16:18,320 --> 00:16:22,320
determine the means of x and y. For example,

214
00:16:22,520 --> 00:16:26,860
suppose I give sum of xi and i goes from 1 to 10

215
00:16:26,860 --> 00:16:31,760
is 700, for example. You have to know that the

216
00:16:31,760 --> 00:16:38,720
sample size is 10, so x bar. is 700 divided by 10,

217
00:16:39,320 --> 00:16:46,180
so it's 7. Then use the curve to compute the

218
00:16:46,180 --> 00:16:52,000
coefficient of correlation. Questions? I think

219
00:16:52,000 --> 00:16:55,900
straightforward, maybe the easiest topic in this

220
00:16:55,900 --> 00:17:02,980
book is to compute the coefficient of correlation

221
00:17:02,980 --> 00:17:09,070
or correlation coefficient. Now my question is, do

222
00:17:09,070 --> 00:17:13,090
you think outliers affect the correlation

223
00:17:13,090 --> 00:17:13,690
coefficient?

224
00:17:17,010 --> 00:17:23,210
We said last time outliers affect the mean, the

225
00:17:23,210 --> 00:17:28,310
range, the variance. Now the question is, do

226
00:17:28,310 --> 00:17:33,510
outliers affect the correlation?

227
00:17:37,410 --> 00:17:38,170
Y.

228
00:17:43,830 --> 00:17:51,330
Exactly. The formula for R has X bar in it or Y

229
00:17:51,330 --> 00:17:56,670
bar. So it means outliers affect

230
00:17:56,670 --> 00:18:01,210
the correlation coefficient. So the answer is yes.

231
00:18:03,470 --> 00:18:06,410
Here we have x bar and y bar. Also, there is

232
00:18:06,410 --> 00:18:10,690
another formula to compute R. That formula is

233
00:18:10,690 --> 00:18:13,370
given by covariance between x and y.

234
00:18:17,510 --> 00:18:21,930
These two formulas are quite similar. I mean, by

235
00:18:21,930 --> 00:18:26,070
using this one, we can end with this formula. So

236
00:18:26,070 --> 00:18:33,090
this formula depends on this x is y. standard

237
00:18:33,090 --> 00:18:36,170
deviations of X and Y. That means outlier will

238
00:18:36,170 --> 00:18:42,530
affect the correlation coefficient. So in case of

239
00:18:42,530 --> 00:18:45,670
outliers, R could be changed.

240
00:18:51,170 --> 00:18:55,530
That formula is called simple correlation

241
00:18:55,530 --> 00:18:58,790
coefficient. On the other hand, we have population

242
00:18:58,790 --> 00:19:02,200
correlation coefficient. If you remember last

243
00:19:02,200 --> 00:19:08,940
time, we used X bar as the sample mean and mu as

244
00:19:08,940 --> 00:19:14,460
population mean. Also, S square as sample variance

245
00:19:14,460 --> 00:19:18,740
and sigma square as population variance. Here, R

246
00:19:18,740 --> 00:19:24,360
is used as sample coefficient of correlation and

247
00:19:24,360 --> 00:19:29,420
rho, this Greek letter pronounced as rho. Rho is

248
00:19:29,420 --> 00:19:35,160
used for population coefficient of correlation.

249
00:19:37,640 --> 00:19:42,040
There are some features of R or Rho. The first one

250
00:19:42,040 --> 00:19:47,960
is unity-free. R or Rho is unity-free. That means

251
00:19:47,960 --> 00:19:54,900
if X represents...

252
00:19:54,900 --> 00:19:58,960
And let's assume that the correlation between X

253
00:19:58,960 --> 00:20:02,040
and Y equals 0.75.

254
00:20:04,680 --> 00:20:07,260
Now, in this case, there is no unity. You cannot

255
00:20:07,260 --> 00:20:13,480
say 0.75 years or 0.75 kilograms. It's unity-free.

256
00:20:13,940 --> 00:20:17,840
There is no unit for the correlation coefficient,

257
00:20:18,020 --> 00:20:21,120
the same as Cv. If you remember Cv, the

258
00:20:21,120 --> 00:20:24,320
coefficient of correlation, also this one is unity

259
00:20:24,320 --> 00:20:30,500
-free. The second feature of R ranges between

260
00:20:30,500 --> 00:20:36,740
minus one and plus one. As I mentioned, R lies

261
00:20:36,740 --> 00:20:42,340
between minus one and plus one. Now, by using the

262
00:20:42,340 --> 00:20:48,100
value of R, you can determine two things. Number

263
00:20:48,100 --> 00:20:53,360
one, we can determine the direction. and strength

264
00:20:53,360 --> 00:20:56,940
by using the sign you can determine if there

265
00:20:56,940 --> 00:21:03,980
exists positive or negative so sign of R determine

266
00:21:03,980 --> 00:21:08,040
negative or positive relationship the direction

267
00:21:08,040 --> 00:21:17,840
the absolute value of R I mean absolute of R I

268
00:21:17,840 --> 00:21:21,980
mean ignore the sign So the absolute value of R

269
00:21:21,980 --> 00:21:24,100
determines the strength.

270
00:21:27,700 --> 00:21:30,760
So by using the sine of R, you can determine the

271
00:21:30,760 --> 00:21:35,680
direction, either positive or negative. By using

272
00:21:35,680 --> 00:21:37,740
the absolute value, you can determine the

273
00:21:37,740 --> 00:21:43,500
strength. We can split the strength into two

274
00:21:43,500 --> 00:21:52,810
parts, either strong, moderate, or weak. So weak,

275
00:21:53,770 --> 00:21:59,130
moderate, and strong by using the absolute value

276
00:21:59,130 --> 00:22:03,870
of R. The closer to minus one, if R is close to

277
00:22:03,870 --> 00:22:07,010
minus one, the stronger the negative relationship

278
00:22:07,010 --> 00:22:09,430
between X and Y. For example, imagine

279
00:22:22,670 --> 00:22:26,130
And as we mentioned, R ranges between minus 1 and

280
00:22:26,130 --> 00:22:26,630
plus 1.

281
00:22:30,070 --> 00:22:35,710
So if R is close to minus 1, it's a strong

282
00:22:35,710 --> 00:22:41,250
relationship. Strong linked relationship. The

283
00:22:41,250 --> 00:22:45,190
closer to 1, the stronger the positive

284
00:22:45,190 --> 00:22:49,230
relationship. I mean, if R is close. Strong

285
00:22:49,230 --> 00:22:54,480
positive. So strong in either direction, either to

286
00:22:54,480 --> 00:22:57,640
the left side or to the right side. Strong

287
00:22:57,640 --> 00:23:00,280
negative. On the other hand, there exists strong

288
00:23:00,280 --> 00:23:05,940
negative relationship. Positive. Positive. If R is

289
00:23:05,940 --> 00:23:10,640
close to zero, weak. Here we can say there exists

290
00:23:10,640 --> 00:23:15,940
weak relationship between X and Y.

291
00:23:19,260 --> 00:23:25,480
If R is close to 0.5 or

292
00:23:25,480 --> 00:23:32,320
minus 0.5, you can say there exists positive

293
00:23:32,320 --> 00:23:38,840
-moderate or negative-moderate relationship. So

294
00:23:38,840 --> 00:23:42,200
you can split or you can divide the strength of

295
00:23:42,200 --> 00:23:44,540
the relationship between X and Y into three parts.

296
00:23:45,860 --> 00:23:50,700
Strong, close to minus one of Plus one, weak,

297
00:23:51,060 --> 00:23:59,580
close to zero, moderate, close to 0.5. 0.5 is

298
00:23:59,580 --> 00:24:04,580
halfway between 0 and 1, and minus 0.5 is also

299
00:24:04,580 --> 00:24:09,040
halfway between minus 1 and 0. Now for example,

300
00:24:09,920 --> 00:24:15,580
what's about if R equals minus 0.5? Suppose R1 is

301
00:24:15,580 --> 00:24:16,500
minus 0.5.

302
00:24:20,180 --> 00:24:27,400
strong negative or equal minus point eight strong

303
00:24:27,400 --> 00:24:33,540
negative which is more strong nine nine because

304
00:24:33,540 --> 00:24:39,670
this value is close closer to minus one than Minus

305
00:24:39,670 --> 00:24:44,070
0.8. Even this value is greater than minus 0.9,

306
00:24:44,530 --> 00:24:50,870
but minus 0.9 is close to minus 1, more closer to

307
00:24:50,870 --> 00:24:56,910
minus 1 than minus 0.8. On the other hand, if R

308
00:24:56,910 --> 00:25:01,190
equals 0.75, that means there exists positive

309
00:25:01,190 --> 00:25:06,970
relationship. If R equals 0.85, also there exists

310
00:25:06,970 --> 00:25:13,540
positive. But R2 is stronger than R1, because 0.85

311
00:25:13,540 --> 00:25:20,980
is closer to plus 1 than 0.7. So we can say that

312
00:25:20,980 --> 00:25:23,960
there exists strong relationship between X and Y,

313
00:25:24,020 --> 00:25:27,260
and this relationship is positive. So again, by

314
00:25:27,260 --> 00:25:32,530
using the sign, you can tell the direction. The

315
00:25:32,530 --> 00:25:35,910
absolute value can tell the strength of the

316
00:25:35,910 --> 00:25:39,870
relationship between X and Y. So there are five

317
00:25:39,870 --> 00:25:44,150
features of R, unity-free. Ranges between minus

318
00:25:44,150 --> 00:25:47,750
one and plus one. Closer to minus one, it means

319
00:25:47,750 --> 00:25:51,950
stronger negative. Closer to plus one, stronger

320
00:25:51,950 --> 00:25:56,410
positive. Close to zero, it means there is no

321
00:25:56,410 --> 00:26:00,790
relationship. Or the weaker, the relationship

322
00:26:00,790 --> 00:26:13,240
between X and Y. By using scatter plots, we

323
00:26:13,240 --> 00:26:18,160
can construct a scatter plot by plotting the Y

324
00:26:18,160 --> 00:26:24,060
values versus the X values. Y in the vertical axis

325
00:26:24,060 --> 00:26:28,400
and X in the horizontal axis. If you look

326
00:26:28,400 --> 00:26:34,500
carefully at graph number one and three, We see

327
00:26:34,500 --> 00:26:42,540
that all the points lie on the straight line,

328
00:26:44,060 --> 00:26:48,880
either this way or the other way. If all the

329
00:26:48,880 --> 00:26:52,320
points lie on the straight line, it means they

330
00:26:52,320 --> 00:26:56,970
exist perfectly. not even strong it's perfect

331
00:26:56,970 --> 00:27:02,710
relationship either negative or positive so this

332
00:27:02,710 --> 00:27:07,530
one perfect negative negative

333
00:27:07,530 --> 00:27:14,090
because x increases y goes down decline so if x is

334
00:27:14,090 --> 00:27:19,590
for example five maybe y is supposed to twenty if

335
00:27:19,590 --> 00:27:25,510
x increased to seven maybe y is fifteen So if X

336
00:27:25,510 --> 00:27:29,290
increases, in this case, Y declines or decreases,

337
00:27:29,850 --> 00:27:34,290
it means there exists negative relationship. On

338
00:27:34,290 --> 00:27:40,970
the other hand, the left corner here, positive

339
00:27:40,970 --> 00:27:44,710
relationship, because X increases, Y also goes up.

340
00:27:45,970 --> 00:27:48,990
And perfect, because all the points lie on the

341
00:27:48,990 --> 00:27:52,110
straight line. So it's perfect, positive, perfect,

342
00:27:52,250 --> 00:27:57,350
negative relationship. So it's straightforward to

343
00:27:57,350 --> 00:27:59,550
determine if it's perfect by using scatterplot.

344
00:28:02,230 --> 00:28:04,950
Also, by scatterplot, you can tell the direction

345
00:28:04,950 --> 00:28:09,270
of the relationship. For the second scatterplot,

346
00:28:09,630 --> 00:28:12,270
it seems to be that there exists negative

347
00:28:12,270 --> 00:28:13,730
relationship between X and Y.

348
00:28:16,850 --> 00:28:21,030
In this one, also there exists a relationship

349
00:28:24,730 --> 00:28:32,170
positive which one is strong more strong this

350
00:28:32,170 --> 00:28:37,110
one is stronger because the points are close to

351
00:28:37,110 --> 00:28:40,710
the straight line much more than the other scatter

352
00:28:40,710 --> 00:28:43,410
plot so you can say there exists negative

353
00:28:43,410 --> 00:28:45,810
relationship but that one is stronger than the

354
00:28:45,810 --> 00:28:49,550
other one this one is positive but the points are

355
00:28:49,550 --> 00:28:55,400
scattered around the straight line so you can tell

356
00:28:55,400 --> 00:29:00,000
the direction and sometimes sometimes not all the

357
00:29:00,000 --> 00:29:04,640
time you can tell the strength sometimes it's very

358
00:29:04,640 --> 00:29:07,960
clear that the relation is strong if the points

359
00:29:07,960 --> 00:29:11,480
are very close straight line that means the

360
00:29:11,480 --> 00:29:15,940
relation is strong now the other one the last one

361
00:29:15,940 --> 00:29:23,850
here As X increases, Y stays at the same value.

362
00:29:23,970 --> 00:29:29,450
For example, if Y is 20 and X is 1. X is 1, Y is

363
00:29:29,450 --> 00:29:33,870
20. X increases to 2, for example. Y is still 20.

364
00:29:34,650 --> 00:29:37,230
So that means there is no relationship between X

365
00:29:37,230 --> 00:29:41,830
and Y. It's a constant. Y equals a constant value.

366
00:29:42,690 --> 00:29:50,490
Whatever X is, Y will have constant value. So that

367
00:29:50,490 --> 00:29:54,790
means there is no relationship between X and Y.

368
00:29:56,490 --> 00:30:01,850
Let's see how can we compute the correlation

369
00:30:01,850 --> 00:30:07,530
between two variables. For example, suppose we

370
00:30:07,530 --> 00:30:12,150
have data for father's height and son's height.

371
00:30:13,370 --> 00:30:16,510
And we are interested to see if father's height

372
00:30:16,510 --> 00:30:21,730
affects his son's height. So we have data for 10

373
00:30:21,730 --> 00:30:28,610
observations here. Father number one, his height

374
00:30:28,610 --> 00:30:38,570
is 64 inches. And you know that inch equals 2

375
00:30:38,570 --> 00:30:39,230
.5.

376
00:30:43,520 --> 00:30:52,920
So X is 64, Sun's height is 65. X is 68, Sun's

377
00:30:52,920 --> 00:30:58,820
height is 67 and so on. Sometimes, if the dataset

378
00:30:58,820 --> 00:31:02,600
is small enough, as in this example, we have just

379
00:31:02,600 --> 00:31:08,640
10 observations, you can tell the direction. I

380
00:31:08,640 --> 00:31:12,060
mean, you can say, yes, for this specific example,

381
00:31:12,580 --> 00:31:15,280
there exists positive relationship between x and

382
00:31:15,280 --> 00:31:20,820
y. But if the data set is large, it's very hard to

383
00:31:20,820 --> 00:31:22,620
figure out if the relation is positive or

384
00:31:22,620 --> 00:31:26,400
negative. So we have to find or to compute the

385
00:31:26,400 --> 00:31:29,700
coefficient of correlation in order to see there

386
00:31:29,700 --> 00:31:32,940
exists positive, negative, strong, weak, or

387
00:31:32,940 --> 00:31:37,820
moderate. but again you can tell from this simple

388
00:31:37,820 --> 00:31:40,280
example yes there is a positive relationship

389
00:31:40,280 --> 00:31:44,660
because just if you pick random numbers here for

390
00:31:44,660 --> 00:31:49,240
example 64 father's height his son's height 65 if

391
00:31:49,240 --> 00:31:54,600
we move up here to 70 for father's height his

392
00:31:54,600 --> 00:32:00,160
son's height is going to be 72 so as father

393
00:32:00,160 --> 00:32:05,020
heights increases Also, son's height increases.

394
00:32:06,320 --> 00:32:11,700
For example, 77, father's height. His son's height

395
00:32:11,700 --> 00:32:15,160
is 76. So that means there exists positive

396
00:32:15,160 --> 00:32:19,740
relationship. Make sense? But again, for large

397
00:32:19,740 --> 00:32:20,780
data, you cannot tell that.

398
00:32:31,710 --> 00:32:36,090
If, again, by using this data, small data, you can

399
00:32:36,090 --> 00:32:40,730
determine just the length, the strength, I'm

400
00:32:40,730 --> 00:32:43,490
sorry, the strength of a relationship or the

401
00:32:43,490 --> 00:32:47,590
direction of the relationship. Just pick any

402
00:32:47,590 --> 00:32:51,030
number at random. For example, if we pick this

403
00:32:51,030 --> 00:32:51,290
number.

404
00:32:55,050 --> 00:33:00,180
Father's height is 68, his son's height is 70. Now

405
00:33:00,180 --> 00:33:02,180
suppose we pick another number that is greater

406
00:33:02,180 --> 00:33:05,840
than 68, then let's see what will happen. For

407
00:33:05,840 --> 00:33:11,060
father's height 70, his son's height increases up

408
00:33:11,060 --> 00:33:17,160
to 72. Similarly, 72 father's height, his son's

409
00:33:17,160 --> 00:33:22,060
height 75. So that means X increases, Y also

410
00:33:22,060 --> 00:33:25,740
increases. So that means there exists both of

411
00:33:25,740 --> 00:33:32,570
them. For sure it is hard to tell this direction

412
00:33:32,570 --> 00:33:36,130
if the data is large. Because maybe you will find

413
00:33:36,130 --> 00:33:40,250
as X increases for one point, Y maybe decreases

414
00:33:40,250 --> 00:33:43,610
for that point. So it depends on the data you

415
00:33:43,610 --> 00:33:49,010
have. Anyway, let's see how can we compute R. I

416
00:33:49,010 --> 00:33:53,770
will use Excel to show how can we do these

417
00:33:53,770 --> 00:33:54,550
calculations.

418
00:34:02,110 --> 00:34:06,530
The screen is clear. But give me the data of X and

419
00:34:06,530 --> 00:34:06,750
Y.

420
00:34:10,710 --> 00:34:14,310
X is 64. 68.

421
00:34:18,910 --> 00:34:26,830
68. 78. There is one 68. 78. 74.

422
00:34:31,120 --> 00:34:37,600
Seventy-four. Seventy-five. Seventy-six.

423
00:34:38,360 --> 00:34:42,240
Seventy-seven. Seventy-five. So that's the values

424
00:34:42,240 --> 00:34:49,440
of X, Y values. Seventy. Seventy-five. Seventy

425
00:34:49,440 --> 00:34:49,800
-seven.

426
00:35:17,230 --> 00:35:23,730
So first we have to compute it. x times y so

427
00:35:23,730 --> 00:35:28,270
that's as x times

428
00:35:28,270 --> 00:35:38,230
the value of y so 46 times 65 equals 4160 x

429
00:35:38,230 --> 00:35:46,050
squared so this value squared for y squared 65

430
00:35:48,660 --> 00:35:53,700
Square and we have to do this one for the rest of

431
00:35:53,700 --> 00:36:03,160
the data So

432
00:36:03,160 --> 00:36:07,600
that's the sum of XY sum X squared and Y squared

433
00:36:07,600 --> 00:36:13,480
now the summation So

434
00:36:13,480 --> 00:36:17,180
that's the sum of X and Y

435
00:36:20,380 --> 00:36:27,040
We have to compute the mean of x and y. So that is

436
00:36:27,040 --> 00:36:31,380
this sum divided by n, where n is 10 in this case.

437
00:36:34,600 --> 00:36:36,100
So this is the first step.

438
00:36:41,820 --> 00:36:48,900
Let's see how can we compute R. R, we have sum of

439
00:36:48,900 --> 00:37:00,780
x, y. minus n is 10 times x bar times y bar. This

440
00:37:00,780 --> 00:37:03,100
is the first quantity. The other one is square

441
00:37:03,100 --> 00:37:08,580
root of sum

442
00:37:08,580 --> 00:37:15,420
x squared minus n x bar squared.

443
00:37:18,830 --> 00:37:24,770
times some y squared minus n times y bar squared.

444
00:37:28,930 --> 00:37:34,090
And we have to find the square root of this value.

445
00:37:34,210 --> 00:37:40,810
So square root, that will give this result. So now

446
00:37:40,810 --> 00:37:46,990
R equals this value divided by

447
00:37:49,670 --> 00:37:54,890
155 and round always to two decimal places will

448
00:37:54,890 --> 00:38:05,590
give 87 so r is 87 so first step we have x and y

449
00:38:05,590 --> 00:38:12,470
compute xy x squared y squared sum of these all of

450
00:38:12,470 --> 00:38:18,100
these then x bar y bar values are given Then just

451
00:38:18,100 --> 00:38:20,820
use the formula you have, we'll get R to be at

452
00:38:20,820 --> 00:38:31,540
seven. So in this case, if we just go back to

453
00:38:31,540 --> 00:38:33,400
the slide we have here.

454
00:38:36,440 --> 00:38:41,380
As we mentioned, father's height is the

455
00:38:41,380 --> 00:38:45,640
explanatory variable. Son's height is the response

456
00:38:45,640 --> 00:38:46,060
variable.

457
00:38:49,190 --> 00:38:52,810
And that simple calculation gives summation of xi,

458
00:38:54,050 --> 00:38:57,810
summation of yi, summation x squared, y squared,

459
00:38:57,970 --> 00:39:02,690
and some xy. And finally, we'll get that result,

460
00:39:02,850 --> 00:39:07,850
87%. Now, the sign is positive. That means there

461
00:39:07,850 --> 00:39:13,960
exists positive. And 0.87 is close to 1. That

462
00:39:13,960 --> 00:39:17,320
means there exists strong positive relationship

463
00:39:17,320 --> 00:39:22,480
between father's and son's height. I think the

464
00:39:22,480 --> 00:39:25,060
calculation is straightforward.

465
00:39:27,280 --> 00:39:33,280
Now, for this example, the data are given in

466
00:39:33,280 --> 00:39:37,460
inches. I mean father's and son's height in inch.

467
00:39:38,730 --> 00:39:41,050
Suppose we want to convert from inch to

468
00:39:41,050 --> 00:39:44,750
centimeter, so we have to multiply by 2. Do you

469
00:39:44,750 --> 00:39:52,050
think in this case, R will change? So if we add or

470
00:39:52,050 --> 00:39:59,910
multiply or divide, R will not change? I mean, if

471
00:39:59,910 --> 00:40:06,880
we have X values, And we divide or multiply X, I

472
00:40:06,880 --> 00:40:09,460
mean each value of X, by a number, by a fixed

473
00:40:09,460 --> 00:40:12,600
value. For example, suppose here we multiplied

474
00:40:12,600 --> 00:40:19,460
each value by 2.5 for X. Also multiply Y by the

475
00:40:19,460 --> 00:40:24,520
same value, 2.5. Y will be the same. In addition

476
00:40:24,520 --> 00:40:28,920
to that, if we multiply X by 2.5, for example, and

477
00:40:28,920 --> 00:40:34,960
Y by 5, also R will not change. But you have to be

478
00:40:34,960 --> 00:40:39,400
careful. We multiply each value of x by the same

479
00:40:39,400 --> 00:40:45,700
number. And each value of y by the same number,

480
00:40:45,820 --> 00:40:49,640
that number may be different from x. So I mean

481
00:40:49,640 --> 00:40:56,540
multiply x by 2.5 and y by minus 1 or plus 2 or

482
00:40:56,540 --> 00:41:01,000
whatever you have. But if it's negative, then

483
00:41:01,000 --> 00:41:05,640
we'll get negative answer. I mean if Y is

484
00:41:05,640 --> 00:41:08,060
positive, for example, and we multiply each value

485
00:41:08,060 --> 00:41:13,000
Y by minus one, that will give negative sign. But

486
00:41:13,000 --> 00:41:17,640
here I meant if we multiply this value by positive

487
00:41:17,640 --> 00:41:21,320
sign, plus two, plus three, and let's see how can

488
00:41:21,320 --> 00:41:22,540
we do that by Excel.

489
00:41:26,320 --> 00:41:31,480
Now this is the data we have. I just make copy.

490
00:41:37,730 --> 00:41:45,190
I will multiply each value X by 2.5. And I will do

491
00:41:45,190 --> 00:41:49,590
the same for Y

492
00:41:49,590 --> 00:41:57,190
value. I will replace this data by the new one.

493
00:41:58,070 --> 00:42:00,410
For sure the calculations will, the computations

494
00:42:00,410 --> 00:42:09,740
here will change, but R will stay the same. So

495
00:42:09,740 --> 00:42:14,620
here we multiply each x by 2.5 and the same for y.

496
00:42:15,540 --> 00:42:19,400
The calculations here are different. We have

497
00:42:19,400 --> 00:42:22,960
different sum, different sum of x, sum of y and so

498
00:42:22,960 --> 00:42:31,040
on, but are the same. Let's see if we multiply

499
00:42:31,040 --> 00:42:38,880
just x by 2.5 and y the same.

500
00:42:41,840 --> 00:42:49,360
So we multiplied x by 2.5 and we keep it make

501
00:42:49,360 --> 00:42:57,840
sense? Now let's see how outliers will affect the

502
00:42:57,840 --> 00:43:03,260
value of R. Let's say if we change one point in

503
00:43:03,260 --> 00:43:08,480
the data set support. I just changed 64.

504
00:43:13,750 --> 00:43:24,350
for example if just by typo and just enter 6 so it

505
00:43:24,350 --> 00:43:33,510
was 87 it becomes 0.7 so there is a big difference

506
00:43:33,510 --> 00:43:38,670
between 0.87 and 0.7 and just we change one value

507
00:43:38,670 --> 00:43:45,920
now suppose the other one is zero 82. The other is

508
00:43:45,920 --> 00:43:48,260
2, for example. 1.

509
00:43:53,380 --> 00:43:59,200
I just changed half of the data. Now R was 87, it

510
00:43:59,200 --> 00:44:02,920
becomes 59. That means these outliers, these

511
00:44:02,920 --> 00:44:06,180
values for sure are outliers and they fit the

512
00:44:06,180 --> 00:44:07,060
correlation coefficient.

513
00:44:11,110 --> 00:44:14,970
Now let's see if we just change this 1 to be 200.

514
00:44:15,870 --> 00:44:20,430
It will go from 50 to up to 63. That means any

515
00:44:20,430 --> 00:44:26,010
changes in x or y values will change the y. But if

516
00:44:26,010 --> 00:44:30,070
we add or multiply all the values by a constant, r

517
00:44:30,070 --> 00:44:31,170
will stay the same.

518
00:44:35,250 --> 00:44:43,590
Any questions? That's the end of chapter 3. I will

519
00:44:43,590 --> 00:44:48,990
move quickly to the practice problems we have. And

520
00:44:48,990 --> 00:44:55,270
we posted the practice in the course website.