XYZ Mat 120 Module 1 Lesson 1 7 — Transcript

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1. 0:00this is module 1 lesson 1.7
2. 0:03and the title is measures of position
3. 0:06we've already looked at measures of
4. 0:07center
5. 0:08measures of variation and this is
6. 0:11measures of
7. 0:12position so what we want to talk about
8. 0:14are a few different things
9. 0:17the first thing we want to talk about is
10. 0:18a z-score a z-score
11. 0:21is a standard score that is the number
12. 0:23of standard deviations a given value
13. 0:26is above or below the mean and we've
14. 0:28kind of talked about
15. 0:30that a little bit before but um when we
16. 0:33go back
17. 0:33and we we look at when we were doing the
18. 0:36empirical rule
19. 0:37we said okay the mean is in here how
20. 0:40many standard deviations we said well
21. 0:42one standard deviation two standard
22. 0:44deviations
23. 0:45three and then back here is a minus one
24. 0:48a minus two and a minus three
25. 0:49those become our z scores so that's what
26. 0:52we're talking about when we're looking
27. 0:54at a z-score
28. 0:55it's simply a formula you take the value
29. 0:58that you're given
30. 0:59subtract the mean and then divide that
31. 1:01quantity by
32. 1:02the standard deviation and so the
33. 1:04population remember we've got x
34. 1:06minus mu divided by sigma so it's the
35. 1:09same calculation
36. 1:10it's just written in the different
37. 1:12formats
38. 1:13so let's look at this problem a normal
39. 1:15distribution
40. 1:17has a mean of 123 a standard deviation
41. 1:21of 5.
42. 1:21we want to find the z-score for a data
43. 1:23value 136.
44. 1:25well my formula for the z-score
45. 1:28is your data value minus the mean
46. 1:33divided by the standard deviation
47. 1:36so we'll do the calculation there we'll
48. 1:38do 136.
49. 1:40so we're going to get rid of this here
50. 1:43and we're going to do 136
51. 1:46we're going to subtract 123 and we want
52. 1:48to get that quantity
53. 1:49that quantity is 13. so it's 13
54. 1:52divided by 5. so 13 divided by 5
55. 1:57so z equals 13 divided by
56. 2:015 and then that gave me a value of 2.6
57. 2:05so 2.6 is the z-score it said 2 decimal
58. 2:09places but that was just
59. 2:102.6 and that's all we need for that
60. 2:14then when we talk about z-scores
61. 2:17generally
62. 2:18they go up to 2 and negative 2.
63. 2:22those are significant values so z scores
64. 2:25are very small
65. 2:26because remember when we talked about
66. 2:28the empirical rule we said that 99.7
67. 2:32of the values fall within 3 standard
68. 2:36deviations so we're not going to get
69. 2:37very high when we're doing z scores
70. 2:40so to talk about significance they're
71. 2:42significant
72. 2:44if they're between two and negative two
73. 2:46otherwise they're significantly high
74. 2:49and they could go further here and
75. 2:50they're significantly low for going to
76. 2:52the left
77. 2:54so for this one we have a recent quiz
78. 2:57the class
79. 2:58mean is 78 standard deviation is 2.7
80. 3:02we want to calculate the z-score if a
81. 3:04person scored 76
82. 3:06so once again we'll say a person's score
83. 3:10minus the mean divided by the standard
84. 3:13deviation
85. 3:15so if i do this calculation
86. 3:18i know that that's a negative 2 on top
87. 3:22divided by 2.7 so
88. 3:25negative 2 divided by 2.7
89. 3:30gives me negative point seven four two
90. 3:33decimal places would be negative
91. 3:35point seven four so equals
92. 3:38negative point seven four and that's
93. 3:40okay to have a negative
94. 3:42because it means that the score was
95. 3:44below the mean
96. 3:46the mean was 78 but this
97. 3:49student scored a 76 so since it's below
98. 3:53then obviously it's going to be to the
99. 3:55left because this is my zero
100. 3:57and so it would be negative 0.74 is this
101. 4:01unusual or not unusual it's not unusual
102. 4:05because remember the values can go
103. 4:08between
104. 4:09negative 2 and 2 to be usual
105. 4:13so those are usual values those are not
106. 4:14unusual those are significant
107. 4:16it's anything outside of that range then
108. 4:18it would be unusual
109. 4:21then we're going to do a five number
110. 4:23summary a five number summary contains
111. 4:26the minimum
112. 4:26q1 the median q3 and the maximum
113. 4:30so i want to go to my calculator and i
114. 4:32want to do stat
115. 4:33i want to do enter and then i want these
116. 4:36data values now i've already entered
117. 4:38these to save a little bit of time so
118. 4:39i've got all the values
119. 4:41entered so you take time and you enter
120. 4:44every single data value that you see
121. 4:46on this chart and
122. 4:50when i get them all in there
123. 4:53i should have 18 values
124. 4:567 18 that's right and then we'll do stat
125. 5:00then we'll do calc and then we'll do
126. 5:02enter
127. 5:03now i want only l1 so if you have
128. 5:06anything in the frequency list clear it
129. 5:08we don't want anything there because we
130. 5:10only have one list and then when we
131. 5:12calculate it we're going to get back up
132. 5:14here what we've seen earlier
133. 5:16so if i go down i'll see my five number
134. 5:19summary
135. 5:20notice your minimum is 11 your q1 is 26
136. 5:24your median is 53.5 your q3 is 81 and
137. 5:28your maximum is 88.
138. 5:30so 11
139. 5:3326 53.5
140. 5:3981 and 88 and that's my five number
141. 5:42summary
142. 5:4381 and 88 so that's what we're looking
143. 5:45at
144. 5:47now we take that and we can plot it
145. 5:51so based on this block box plot we want
146. 5:54to identify
147. 5:55a five number summary well let me go
148. 5:57back to this for just a minute
149. 5:59if i go to second y and i do a plot
150. 6:03i want this on and i come down here and
151. 6:06we did this
152. 6:07back in the graphing part at the
153. 6:09beginning of this module
154. 6:11and we do a box plot remember we can go
155. 6:14to zoom and we can go to stat
156. 6:19and it'll show me my picture now that's
157. 6:20not this particular picture that's the
158. 6:22one previous
159. 6:23and so here is your your minimum here's
160. 6:26your q1
161. 6:27here's your median here's your q3 and
162. 6:28then there's your maximum value
163. 6:30so now based on that let's look at this
164. 6:32one the box plot
165. 6:34the minimum that's 10 right here
166. 6:38the next one is this value that's 11.
167. 6:41so that would be q1 here's my median
168. 6:45that would be 12. here's 14 which would
169. 6:48be q3
170. 6:50and then my maximum is over here at 19.
171. 6:53so that's how we read the box plots
172. 6:58based on this box plot where is the
173. 7:00median okay remember
174. 7:02we start with the minimum we have q1
175. 7:05here's the median right here based on
176. 7:07this box plot
177. 7:08the median is 11. q3 is over here and
178. 7:11then the maximum
179. 7:14now we're going to talk about the
180. 7:15interquartile range and this is a
181. 7:17formula
182. 7:18the interquartile range gives us some
183. 7:20boundaries
184. 7:21um we're going to find q1 q2 and q3 and
185. 7:24then here's a rule
186. 7:25the interquartile range is q3 minus q1
187. 7:28so you just subtract those values but
188. 7:31to evaluate this to see if we have any
189. 7:34outliers we're going to take that value
190. 7:36that we get
191. 7:37and multiply by 1.5 anything
192. 7:41greater than q3 would be an outlier
193. 7:45and anything smaller than q1 would be an
194. 7:48outlier
195. 7:49and so what we're doing is we're
196. 7:50bounding and we're figuring out what
197. 7:52numbers
198. 7:52would fall in that interquartile range
199. 7:56so now we have a stem and leaf plot
200. 7:59so let's go back up here and it says
201. 8:01find all this stuff the five number
202. 8:03summary the range the inner quartile
203. 8:05range
204. 8:06so looking at this i'm going to go back
205. 8:09to my calculator
206. 8:11i'm going to put these values in for l1
207. 8:14the first one is 10 the next one is 12.
208. 8:18remember we're reading this just like we
209. 8:21read it when we did our stem and leaf
210. 8:23plot interpretation
211. 8:2524 29 and 29
212. 8:28then we have nothing for three because
213. 8:30there's no zero there so no three
214. 8:33we have a 42 another 42
215. 8:36a 48 and a 49.
216. 8:40and then we've got a 50 because we do
217. 8:42have a 0 there
218. 8:43we have a 52 we have a 53
219. 8:47we have a 54 we have a 58 and we have a
220. 8:5059
221. 8:51and then the last one we do have a 60
222. 8:53because of the zero and we have a 65.
223. 8:56so those are my values i should have 19
224. 8:59values
225. 9:04okay so the minimum we'll go back and
226. 9:07we'll do stat we'll do calc
227. 9:08one variable stats nothing in the
228. 9:10frequency list
229. 9:12and then we'll come down here okay so
230. 9:15the minimum is 10
231. 9:17q1 is 24 and the median is 48.
232. 9:20so 10
233. 9:2424 48 is the median
234. 9:28then we had a 54 and a 65
235. 9:3254 and 65. so that's my inter
236. 9:36not my interquartile that's my five
237. 9:37number summary the range
238. 9:39remember the range is simply
239. 9:44the maximum minus the minimum
240. 9:47so max minus the min so the range is 65
241. 9:50minus 10 i know that's 55.
242. 9:52what is the interquartile range the
243. 9:54interquartile range is just simply
244. 9:58q3 minus q1 so q3
245. 10:02is 54 q1 is 24
246. 10:06so that will give me a interquartile
247. 10:08range of 30.
248. 10:11so that's how i calculate that and then
249. 10:12the very last one
250. 10:14we want to do the same thing and then we
251. 10:16want to figure out the bounds
252. 10:18for our possible outliers
253. 10:21so let's do this one by going to stat
254. 10:24we'll hit our
255. 10:30list
256. 10:32put all these values in here
257. 10:54i'm hoping i'm getting all these right
258. 10:58um so now we want to do the same thing i
259. 11:00did before
260. 11:01stat calc list
261. 11:04we'll scroll down and we're going to
262. 11:07find
263. 11:08the minimum the q1 the q3 and the
264. 11:09maximum so let's look at this 11
265. 11:1137 46 57
266. 11:14so 11 37 46
267. 11:1857 and 85
268. 11:24so those are my values for the five
269. 11:27number summary
270. 11:28what is the value for iqr so remember
271. 11:31that's q3 minus q1
272. 11:34so that would be 57 minus
273. 11:3737 so that would be 27 i'm 20 i'm sorry
274. 11:43so 20 3 minus q1 so that would be 20
275. 11:47and then the lower bound for the outlier
276. 11:50this is where i want to take the 1.5
277. 11:53and multiply it times the interquartile
278. 11:56range
279. 11:57so 20 times 1.5 20
280. 12:00times 1.5 is going to be
281. 12:0430. so that would be my 30
282. 12:08and i want to subtract that from q1
283. 12:13so q1 is 37
284. 12:16so 37 minus 30 is going to give me seven
285. 12:22and then for the upper so
286. 12:25i've got 30 and i want to add it to q3
287. 12:28so i'm going to take 57 and i'm going to
288. 12:30add
289. 12:3030 and i'll get 87. and so we're
290. 12:33expecting our ranges
291. 12:35of these numbers to fall between 7 and
292. 12:3787
293. 12:38anything outside of those range values
294. 12:41would be considered unusual