Z-statistics vs. T-statistics | Inferential statistics | Probability and Statistics | Khan Academy — Transcript

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1. 0:00I want to use this video to kind of make sure we
2. 0:03intuitively and otherwise and understand the difference
3. 0:06between a Z-statistic-- something I have trouble
4. 0:13saying-- and a T-statistic.
5. 0:19So in a lot of what we're doing in this inferential
6. 0:22statistics, we're trying to figure out what is the
7. 0:25probability of getting a certain sample mean.
8. 0:28So what we've been doing, especially when we have a
9. 0:30large sample size-- so let me just draw a sampling
10. 0:34distribution here.
11. 0:35So let's say we have a sampling distribution of the
12. 0:38sample mean right here.
13. 0:40It has some assumed mean value and some standard deviation.
14. 0:48What we want to do is any result that we get, let's say
15. 0:52we get some sample mean out here.
16. 0:55We want to figure out the probability of getting a
17. 0:57result at least as extreme as this.
18. 0:59So you can either figure out the probability of getting a
19. 1:03result below this and subtracted that from 1, or
20. 1:05just figure out this area right over there.
21. 1:08And to do that we've been figuring out how many standard
22. 1:11deviations above the mean we actually are.
23. 1:15The way we figured that out is we take our sample mean, we
24. 1:20subtract from that our mean itself, we subtract from that
25. 1:26what we assume the mean should be, or maybe we don't know
26. 1:28what this is.
27. 1:31And then we divide that by the standard deviation of the
28. 1:36sampling distribution.
29. 1:42This is how many standard deviations we
30. 1:44are above the mean.
31. 1:46That is that distance right over there.
32. 1:48Now, we usually don't know what this is either.
33. 1:52We normally don't know what that is either.
34. 1:54And the central limit theorem told us that assuming that we
35. 2:01have a sufficient sample size, this thing right here, this
36. 2:04thing is going to be the same thing as-- the sample is going
37. 2:09to be the same thing as the standard deviation of our
38. 2:12population divided by the square root
39. 2:17of our sample size.
40. 2:19So this thing right over here can be re-written as our
41. 2:24sample mean minus the mean of our sampling distribution of
42. 2:30the sample mean divided by this thing right here--
43. 2:34divided by our population mean, divided by the square
44. 2:37root of our sample size.
45. 2:39And this is essentially our best sense of how many
46. 2:41standard deviations away from the actual mean we are.
47. 2:45And this thing right here, we've learned it before, is a
48. 2:48Z-score, or when we're dealing with an actual statistic when
49. 2:51it's derived from the sample mean statistic, we call this a
50. 2:55Z-statistic.
51. 2:58And then we could look it up in a Z-table or in a normal
52. 3:02distribution table to say what's the probability of
53. 3:04getting a value of this Z or greater.
54. 3:08So that would give us that probability.
55. 3:09So what's the probability of getting that
56. 3:11extreme of a result?
57. 3:13Now normally when we've done this in the last few videos,
58. 3:18we also do not know what the standard deviation of the
59. 3:24population is.
60. 3:25So in order to approximate that we say that the Z-score
61. 3:31is approximately, or the Z-statistic, is approximately
62. 3:34going to be-- so let me just write the numerator over
63. 3:38again-- over, we estimate this using our sample standard
64. 3:42deviation-- let me do this in a new color-- with using our
65. 3:48sample standard deviation.
66. 3:53And this is OK if our sample size is greater than 30.
67. 4:02Or another way to think about it is this will be normally
68. 4:05distributed if our sample size is greater than 30.
69. 4:14Even this approximation will be approximately normally
70. 4:16distributed.
71. 4:17Now, if your sample size is less than 30, especially if
72. 4:21it's a good bit less than 30, all of a sudden this
73. 4:23expression will not be normally distributed.
74. 4:25So let me re-write the expression over here.
75. 4:28Sample mean minus the mean of your sampling distribution of
76. 4:32the sample mean divided by your sample standard deviation
77. 4:35over the square root of your sample size.
78. 4:39We just said if this thing is well over 30, or at least 30,
79. 4:44then this value right here, this statistic, is going to be
80. 4:48normally distributed.
81. 4:49If it's not, if this is small, then this is going to have a
82. 4:55T-distribution.
83. 5:00And then you're going to do the exact same thing you did
84. 5:02here, but now you would assume that the bell is no longer a
85. 5:04normal distribution, so this example it was normal.
86. 5:09All of Z's are normally distributed.
87. 5:11Over here in a T-distribution, and this will actually be a
88. 5:14normalized T-distribution right here because we
89. 5:16subtracted out the mean.
90. 5:18So in a normalized T-distribution, you're going
91. 5:22to have a mean of 0.
92. 5:24And what you're going to do is you want to figure out the
93. 5:26probability of getting a T-value at least this extreme.
94. 5:30So this is your T-value you would get, and then you
95. 5:34essentially figure out the area under the curve right
96. 5:37over there.
97. 5:39So a very easy rule of thumb is calculate this quantity
98. 5:43either way.
99. 5:44Calculate this quantity either way.
100. 5:47If you will have more than 30 samples, if your sample size
101. 5:51is more than 30, your sample standard deviation is going to
102. 5:55be a good approximator for your
103. 5:57population standard deviation.
104. 5:59And so this whole thing is going to be approximately
105. 6:01normally distributed, and so you can use a Z-table to
106. 6:04figure out the probability of getting a result
107. 6:06at least that extreme.
108. 6:08If your sample size is small, then this statistic, this
109. 6:14quantity, is going to have a T-distribution, and then
110. 6:19you're going to have to use a T-table to figure out the
111. 6:22probability of getting a T-value at least this extreme.
112. 6:27And we're going to see this in an example a couple
113. 6:29of videos from now.
114. 6:30Anyway, hopefully that helped clarify some things in your
115. 6:32head about when to use a Z-statistic or when to use a
116. 6:36T-statistic.