Life Tables (Part 1: Theory and Definition) | Actuarial Sciences | Vital Statistics — Transcript

Full transcript

1. 0:03Hello dear friends, this is Dashiish and
2. 0:04I welcome you all to my channel
3. 0:06statistics curiosity. Friends, today we
4. 0:08will learn what is a life table. So a
5. 0:11life table is actually a statistical
6. 0:13tool. It is very important tool by the
7. 0:15way and it is used in demography.
8. 0:17Demography is also popularly known as
9. 0:20population sciences.
10. 0:24Okay, it is also known as population
11. 0:26sciences. Apart from this, it is also
12. 0:29known as uh vital stats,
13. 0:32vital statistics.
14. 0:34Okay. So, it is majorly used in
15. 0:36population science and uh it is also
16. 0:38used in epidemiology and acturial
17. 0:41science. It is also used in acturial
18. 0:43sciences. Actual sciences actually it is
19. 0:45a science related to uh you know
20. 0:48insuranceances.
21. 0:51Okay. Here where a person decides uh you
22. 0:55uh actually the person called acturies
23. 1:00they decide the premium of
24. 1:02insuranceances using the life table. So
25. 1:05life table is very important. It is uh
26. 1:07it is also summarizes the pattern of
27. 1:10mortality and survival in a population.
28. 1:13So when you have entire life table you
29. 1:16can see the pattern of
30. 1:19survival as well as death. Pattern of
31. 1:23survival as well as
32. 1:27death. I should also give you uh a
33. 1:30formal definition following [snorts]
34. 1:32G barley. Garlay is one of the famous uh
35. 1:36statistician.
36. 1:38So he gave a definition of life table.
37. 1:43So as per GL
38. 1:45uh the life table is actually a it is a
39. 1:50life history it is a
40. 1:54life history of
41. 1:57a cohort
42. 1:59as it
43. 2:02diminishes
44. 2:04as it diminishes gradually by deaths.
45. 2:10gradually by deaths.
46. 2:14And uh the record of the record
47. 2:19uh begins
48. 2:21the record begins
49. 2:24at the birth of
50. 2:27At the birth of
51. 2:31each member
52. 2:34and
53. 2:37continues
54. 2:41until all have died.
55. 2:44until all have died.
56. 2:48So actually what happens is you start
57. 2:50with a cohort.
58. 2:53Cohort means a group of people. A group
59. 2:56of people
60. 2:59let's say thousand people you start with
61. 3:02these people. Now these people I mean uh
62. 3:05at they might be at some age. Okay.
63. 3:08Let's say the age x is zero. So at age
64. 3:12zero you have total 1,000 people. Okay.
65. 3:151,000 child. Okay. And at age one these
66. 3:20are now remaining 980. At age two they
67. 3:23are now 950. At age three let's say they
68. 3:27are 938.
69. 3:30So they are keep on decreasing
70. 3:32as uh as as uh as the time passes uh the
71. 3:36the these thousand people uh this uh
72. 3:39this number diminishes gradually as the
73. 3:42definition says okay it decreases
74. 3:45gradually and it become uh finally it
75. 3:48will become zero. So this entire record
76. 3:53this entire record is nothing but this
77. 3:56is called a this is called a life table.
78. 4:01Okay. So life table is nothing but a
79. 4:03record of these deaths.
80. 4:06How many deaths are there? Okay. So you
81. 4:09start with a number that number is
82. 4:10called cohort sometime also called
83. 4:13radics.
84. 4:15And this number keep on decreasing and
85. 4:18uh till it becomes zero. So this record
86. 4:22is called life table. And there are
87. 4:24several other columns of life table
88. 4:26which we will discuss which we will
89. 4:29discuss now. Okay.
90. 4:31Okay. Now uh now come to the components
91. 4:34of life table. So uh life table is
92. 4:37actually uh an entire table with several
93. 4:39columns. So these columns are called
94. 4:43these
95. 4:45uh these columns
96. 4:48these are called components of life
97. 4:50table. Okay. So the first column is
98. 4:52obviously the age. So this age column
99. 4:56might be an interval.
100. 4:58Okay. Or a simple single number. Okay.
101. 5:02So this might be an interval age. A age
102. 5:05interval.
103. 5:08Okay. Or it might be directly a simple
104. 5:12single number. Let's say is 1, is 15 and
105. 5:16so on. So life table generally created
106. 5:19at equal interval of time. And this
107. 5:21might be 0 to 5, 5 to 10 and so on.
108. 5:25Okay. So this is the age interval. So
109. 5:27the table is divided into specific age
110. 5:29groups or single year ages. The single
111. 5:32year a years age age years is often
112. 5:34denoted by X. Okay. Now the second
113. 5:37column is the number alive that is L X L
114. 5:40suffix max. Okay. So it shows how many
115. 5:44people are alive at the beginning of
116. 5:46each each age interval. So this uh
117. 5:50symbol means the it is the number of
118. 5:52persons surviving to the exact age x
119. 5:55years. So if you are writing L15 that
120. 5:59means number of
121. 6:01people's
122. 6:05surviving
123. 6:08till age yeah at age 15.
124. 6:11Okay.
125. 6:14Okay. So uh one more thing if you write
126. 6:17uh uh if you write L0
127. 6:20so it means how many people are uh at
128. 6:24the initial study okay so L0 is the
129. 6:27assumed number of birds not birds it is
130. 6:31birth
131. 6:33assumed number of birds okay that uh we
132. 6:36we were called that as cohort or radics
133. 6:39that we have talked earlier okay so this
134. 6:43number might be 1,000 10,000 1 lakh or
135. 6:45so on. Okay. So this is L0. Okay. So
136. 6:49obviously if you are starting from a
137. 6:51zero so it will be like this L0 L1 L2
138. 6:55and so on. Okay. So now third column is
139. 7:00number D that is dx. So dx indicates how
140. 7:04many people die during a each age
141. 7:06interval. So it is nothing but the how
142. 7:09many people are dying. Let's say at
143. 7:11initial at initial L0 is 1,000 and it is
144. 7:16900. So 1,000 - 900 that means 100
145. 7:20people are dying. Okay. So dx is nothing
146. 7:23but the number of people that die during
147. 7:26that particular age interval or from
148. 7:28that particular age x to x + 1. Okay. So
149. 7:33uh by following this rule you can write
150. 7:36a formula for uh dx
151. 7:39for age group x2 x + 1 for taking a
152. 7:42taking age x
153. 7:45x + 1 x + 2 and so on for this age. So
154. 7:50you will have a formula dx is equals to
155. 7:55l x - lx + 1. Okay. So if this L0 is lx
156. 8:01so lx + 1 will be l1. So you will
157. 8:04substract 1,000 you will subtract 900
158. 8:07from 1,000,000 - 900. So that number 100
159. 8:10will be termed as how many people died.
160. 8:13Okay. Okay. So this is the number time
161. 8:16that is dx. So this is a formula for dx.
162. 8:20Okay. Now come to probability of time
163. 8:23that is qx. So qx is nothing but the
164. 8:26likelihood that an individual of age x
165. 8:28will die before reaching age x + 1. So
166. 8:31it is very simple. It is qx is equals to
167. 8:34dx divided by lx. Okay it is qx = to dx
168. 8:39divided by lx. The number of people
169. 8:41dying divided by total number of people
170. 8:44surviving till that age. Okay. So the
171. 8:47ratio of this will be the ratio of this
172. 8:50both will be your probability of time.
173. 8:53Okay. So that is mortality probability.
174. 8:57So obviously the uh uh the opposite of
175. 9:00this probability will be termed as px.
176. 9:03That is called survival probability.
177. 9:06That is called survival
178. 9:09probability. And this survival
179. 9:11probability will obviously be px = to 1
180. 9:14- qx.
181. 9:17Okay. So this will be px = to 1 - qx.
182. 9:22So this number one, number two, number
183. 9:25three. Now next column is capital LX.
184. 9:29Okay, capital L X. The average
185. 9:31stationary population living between age
186. 9:34X and X + 1. So capital L X is average
187. 9:39stationary population. So
188. 9:42L X and L X + 1 divided by 2. Okay.
189. 9:47So it is average age. So if you are
190. 9:50having l x l x + 1 l x + 2 and so on. So
191. 9:56l x + l x + 1 2 divided by 2. This will
192. 10:00be the average age. This average age
193. 10:03between two uh time intervals. Okay. Or
194. 10:07between two age x and x + 1. Okay.
195. 10:11Now what is uh next column? Next column
196. 10:14is tx capital tx. So TX is the total
197. 10:18number of years lived by the cohort
198. 10:21after attaining an exact age X.
199. 10:25So what is this? TX is actually it is L
200. 10:29X
201. 10:30+ L X + 1 + L X + 2 plus and so on.
202. 10:37Okay. So we will we will also see this
203. 10:40during numerical. It will be clear more
204. 10:43clear during numerical. But now it uh
205. 10:46you can consider it as a formula. So
206. 10:48actually what you are doing you are
207. 10:50creating several uh average ages
208. 10:55at consecutive consecutive time periods
209. 10:58and you are adding those uh adding those
210. 11:01average ages. So actually in other words
211. 11:04you can also say it removes the total
212. 11:07future years lived by small lx persons
213. 11:10of the cohort who have attended the age
214. 11:12x. So what you will do you will actually
215. 11:16you will first you will you will have x
216. 11:18column lx column then you will calculate
217. 11:21capital l x column. So there will be
218. 11:24capital l capital l1 capital l2 and so
219. 11:27on. Then you will compute capital tx
220. 11:30column. So t0 will be what? T0 will be
221. 11:34l0 to last l + l1 + l2 sum of this l0 to
222. 11:39let's say l10. Okay. and then t1 will be
223. 11:42sum of capital L1 till L10. TS2 will be
224. 11:46sum of capital L2 till L10 and so on.
225. 11:49Okay. So this was your capital TX
226. 11:53column. So and this is the formula that
227. 11:55I have mentioned. You can write this
228. 11:57formula on your paper. Okay.
229. 12:02Now next and last column of the life
230. 12:05table which is also very important
231. 12:07column that is called life expectancy.
232. 12:10It is the average number of additional
233. 12:13years a person of age X is expected to
234. 12:16live.
235. 12:18Okay. So what is the formula? It is ex
236. 12:21equals to it is sometimes also denoted
237. 12:23as EX note. In some books it is exe but
238. 12:28we are writing it as EX. So what is ex?
239. 12:32EX is equals to TX / NX.
240. 12:37So ex equals to capital TX / LX. So what
241. 12:41it will give? It will give you average
242. 12:44number of additional years or how many
243. 12:47years you will live after age X. After
244. 12:51age X let's say you have E 10 equals to
245. 12:5567 years.
246. 12:57So after 10 years age you will also live
247. 13:0167 additional years. That is the meaning
248. 13:05of life expectancy. So your total life
249. 13:08uh average life should be 77 67 + 10.
250. 13:12Okay friends, we have seen all the
251. 13:13columns of life table. What are their
252. 13:15meanings and how to compute them? What
253. 13:17are the formulas and we have also seen
254. 13:19the definition of life table. Now in
255. 13:21second part of the video in part two of
256. 13:24the video I will solve a numerical
257. 13:27manually as well as using Excel.
258. 13:32So I recommend you to uh watch that part
259. 13:35two of life table that numerical also to
260. 13:38get full information full confidence uh
261. 13:41in learning the life table. Okay. Thank
262. 13:44you for watching.