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/exercises/standard_deviation.html

https://github.com/beneater/khan-exercises
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  1. <!DOCTYPE html>
    2. 

  2. <html data-require="math math-format word-problems stat">
    3. 

  3. <head>
    4. 

  4. 	<meta charset="UTF-8" />
    5. 

  5. 	<title>Standard deviation</title>
    6. 

  6. 	<script src="../khan-exercise.js"></script>
    7. 

  7. 	<style>
    8. 

  8. 		#answer_area .short input[type=text] {
    9. 

  9. 			width: 40px;
    10. 

  10. 		}
    11. 

  11. 	</style>
    12. 

  12. </head>
    13. 

  13. <body>
    14. 

  14. 	<div class="exercise">
    15. 

  15. 		<div class="vars">
    16. 

  16. 			<var id="DATA_POINTS">randRange( 4, 6 )</var>
    17. 

  17. 			<var id="POPULATION">randRange( 20, 50 )</var>
    18. 

  18. 			<var id="TGT_MEAN">animalAvgLifespan( 1 )</var>
    19. 

  19. 			<var id="TGT_STDDEV">animalStddevLifespan( 1 )</var>
    20. 

  20. 			<var id="DATA">jQuery.map( randGaussian( TGT_MEAN, TGT_STDDEV, DATA_POINTS ), function( lifespan ) {
    21. 

  21. 				lifespan = lifespan &lt; 1 ? 1 : round( lifespan );
    22. 

  22. 				return randRange( 1, lifespan );
    23. 

  23. 			} )</var>
    24. 

  24. 			<var id="MEAN">roundTo( 1, mean( DATA ) )</var>
    25. 

  25. 			<var id="SQR_DEV">jQuery.map( DATA, function( x ) { return roundTo( 2, ( x - MEAN ) * ( x - MEAN ) ); })</var>
    26. 

  26. 			<var id="VARIANCE">roundTo( 2, sum( SQR_DEV ) / ( DATA_POINTS - 1 ) )</var>
    27. 

  27. 			<var id="VARIANCE_POP">roundTo( 2, sum( SQR_DEV ) /  DATA_POINTS )</var>
    28. 

  28. 			<var id="STDDEV">roundTo( 1, stdDev( DATA ) )</var>
    29. 

  29. 			<var id="STDDEV_POP">roundTo( 1, stdDevPop( DATA ) )</var>
    30. 

  30. 		</div> <!-- vars -->
    31. 

  31. 

  32. 		<div class="problems">
    33. 

  33. 

  34. 			<div id="population">
    35. 

  35. 				<div class="problem" data-else>
    36. 

  36. 					<p>You have found the following ages (in years) of all <var>plural( DATA_POINTS, animal( 1 ) )</var> at your local zoo:</p>
    37. 

  37. 					<p><code>\qquad<var>DATA.join( ",\\enspace " )</var></code></p>
    38. 

  38. 				</div>
    39. 

  39. 

  40. 				<p class="question">
    41. 

  41. 					What is the average age of the <var>plural( animal( 1 ) )</var> at your zoo? What is the standard deviation?
    42. 

  42. 					You may round your answers to the nearest tenth.
    43. 

  43. 				</p>
    44. 

  44. 

  45. 				<div class="solution" data-type="multiple">
    46. 

  46. 					<p>
    47. 

  47. 						Average age:
    48. 

  48. 						<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>mean( DATA )</var></span> years old
    49. 

  49. 					</p>
    50. 

  50. 					<p>
    51. 

  51. 						Standard deviation:
    52. 

  52. 						<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>stdDevPop( DATA )</var></span> years
    53. 

  53. 					</p>
    54. 

  54. 					<div class="example">decimals, like <code>7.5</code></div>
    55. 

  55. 					<div class="example">answers within <code>\pm 0.15</code> are accepted to allow for rounding part-way through</div>
    56. 

  56. 				</div> <!-- solution -->
    57. 

  57. 

  58. 				<div class="hints">
    59. 

  59. 					<p>
    60. 

  60. 						Because we have data for all <var>plural( DATA_POINTS, animal( 1 ) )</var> at the zoo, we are able
    61. 

  61. 						to calculate the <span class="hint_blue">population mean</span>
    62. 

  62. 						<code>(\color{<var>BLUE</var>}{\mu})</code> and
    63. 

  63. 						<span class="hint_pink">population standard deviation</span> <code>(\color{<var>PINK</var>}{\sigma})</code>.
    64. 

  64. 					</p>
    65. 

  65. 

  66. 					<div>
    67. 

  67. 						<p>
    68. 

  68. 							To find the <span class="hint_blue">population mean</span>, add up the values of all <code class="hint_green"><var>DATA_POINTS</var></code>
    69. 

  69. 							ages and divide by <code class="hint_green"><var>DATA_POINTS</var></code>.
    70. 

  70. 						</p>
    71. 

  71. 						<p>
    72. 

  72. 							<code>
    73. 

  73. 								\color{<var>BLUE</var>}{\mu} \quad = \quad
    74. 

  74. 								\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{N}} x_i}{\color{<var>GREEN</var>}{N}} \quad = \quad
    75. 

  75. 								\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} x_i}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}}
    76. 

  76. 							</code>
    77. 

  77. 						</p>
    78. 

  78. 					</div>
    79. 

  79. 

  80. 					<p>
    81. 

  81. 						<code>
    82. 

  82. 								\color{<var>BLUE</var>}{\mu} \quad = \quad
    83. 

  83. 								\dfrac{<var>plus.apply( KhanUtil, DATA )</var>}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} \quad = \quad
    84. 

  84. 								\color{<var>BLUE</var>}{<var>MEAN</var>\text{ <var>plural( "year", MEAN )</var> old}}
    85. 

  85. 						</code>
    86. 

  86. 					</p>
    87. 

  87. 

  88. 					<div>
    89. 

  89. 						<p>
    90. 

  90. 							Find the <span class="hint_purple">squared deviations from the mean</span> for each <var>animal(1)</var>.
    91. 

  91. 						</p>
    92. 

  92. 						<div class="fake_header">
    93. 

  93. 							<span style="width: 100px;">
    94. 

  94. 								Age<br/>
    95. 

  95. 								<code>x_i</code>
    96. 

  96. 							</span><span style="width: 150px;">
    97. 

  97. 								<span class="hint_gray">Distance from the mean</span>
    98. 

  98. 								<code>(x_i - \color{<var>BLUE</var>}{\mu})</code>
    99. 

  99. 							</span><span style="width: 150px;">
    100. 

  100. 								<code>(x_i - \color{<var>BLUE</var>}{\mu})^2</code>
    101. 

  101. 							</span>
    102. 

  102. 						</div>
    103. 

  103. 						<div class="fake_row" data-each="DATA as i, POINT">
    104. 

  104. 							<span style="width: 100px;">
    105. 

  105. 								<code><var>POINT</var></code> <var>plural( "year", POINT )</var>
    106. 

  106. 							</span><span style="width: 150px;" class="hint_gray">
    107. 

  107. 								<code><var>roundTo( 2, POINT - MEAN )</var></code> <var>plural( "year", roundTo( 2, POINT - MEAN ) )</var>
    108. 

  108. 							</span><span style="width: 150px;" class="hint_purple">
    109. 

  109. 								<code><var>SQR_DEV[ i ]</var></code> <var>plural( "year", SQR_DEV[ i ] )</var><code>^2</code>
    110. 

  110. 							</span>
    111. 

  111. 						</div>
    112. 

  112. 					</div>
    113. 

  113. 

  114. 					<div>
    115. 

  115. 						<p>
    116. 

  116. 							Because we used the <span class="hint_blue">population mean</span><code>(\color{<var>BLUE</var>}{\mu})</code> to compute the
    117. 

  117. 							<span class="hint_purple">squared deviations from the mean</span>, we can find the <span class="hint_red">variance</span>
    118. 

  118. 							<code>(\color{red}{\sigma^2})</code>, without introducing any bias, by simply averaging the
    119. 

  119. 							<span class="hint_purple">squared deviations from the mean</span>:
    120. 

  120. 						</p>
    121. 

  121. 						<p>
    122. 

  122. 							<code>
    123. 

  123. 								\color{red}{\sigma^2} \quad = \quad
    124. 

  124. 								\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{N}} (x_i - \color{<var>BLUE</var>}{\mu})^2}{\color{<var>GREEN</var>}{N}}
    125. 

  125. 							</code>
    126. 

  126. 						</p>
    127. 

  127. 					</div>
    128. 

  128. 

  129. 					<p>
    130. 

  130. 						<code>
    131. 

  131. 							\color{red}{\sigma^2} \quad = \quad
    132. 

  132. 							\dfrac{<var>plus.apply( KhanUtil, jQuery.map( SQR_DEV, function( x ) { return "\\color{purple}{" + x + "}"; }) )</var>}
    133. 

  133. 							{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}}
    134. 

  134. 						</code>
    135. 

  135. 					</p>
    136. 

  136. 

  137. 					<p>
    138. 

  138. 						<code>
    139. 

  139. 							\color{red}{\sigma^2} \quad = \quad
    140. 

  140. 							\dfrac{\color{purple}{<var>roundTo( 2, sum( SQR_DEV ) )</var>}}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} \quad = \quad
    141. 

  141. 							\color{red}{<var>VARIANCE_POP</var>\text{ <var>plural( "year", VARIANCE_POP )</var>}^2}
    142. 

  142. 						</code>
    143. 

  143. 					</p>
    144. 

  144. 

  145. 					<div>
    146. 

  146. 						<p>
    147. 

  147. 							As you might guess from the notation, the <span class="hint_pink">population standard deviation</span>
    148. 

  148. 							<code>(\color{<var>PINK</var>}{\sigma})</code> is found by taking the square root of the <span class="hint_red">population variance</span>
    149. 

  149. 							<code>(\color{red}{\sigma^2})</code>.
    150. 

  150. 						</p>
    151. 

  151. 						<p>
    152. 

  152. 							<code>\color{<var>PINK</var>}{\sigma} = \sqrt{\color{red}{\sigma^2}}</code>
    153. 

  153. 						</p>
    154. 

  154. 					</div>
    155. 

  155. 

  156. 					<p>
    157. 

  157. 						<code>
    158. 

  158. 							\color{<var>PINK</var>}{\sigma} = \sqrt{\color{red}{<var>VARIANCE_POP</var>\text{ <var>plural( "year", VARIANCE_POP )</var>}^2}} =
    159. 

  159. 							\color{<var>PINK</var>}{<var>STDDEV_POP</var>\text{ <var>plural( "year", STDDEV_POP )</var>}}
    160. 

  160. 						</code>
    161. 

  161. 					</p>
    162. 

  162. 

  163. 					<p><strong>
    164. 

  164. 						The average <var>animal( 1 )</var> at the zoo is <var>plural( MEAN, "year" )</var> old with a standard deviation
    165. 

  165. 						of <var>plural( STDDEV_POP, "year" )</var>.
    166. 

  166. 					</strong></p>
    167. 

  167. 

  168. 				</div> <!-- hints -->
    169. 

  169. 			</div> <!-- population -->
    170. 

  170. 

  171. 			<div id="sample">
    172. 

  172. 				<div class="problem" data-else>
    173. 

  173. 					<p>
    174. 

  174. 						You have found the following ages (in years) of <var>plural( DATA_POINTS, animal( 1 ) )</var>
    175. 

  175. 						randomly selected from the <var>plural( POPULATION, animal( 1 ) )</var> at your local zoo:
    176. 

  176. 					</p>
    177. 

  177. 					<p><code>\qquad<var>DATA.join( ",\\enspace " )</var></code></p>
    178. 

  178. 				</div>
    179. 

  179. 

  180. 				<p class="question">
    181. 

  181. 					Based on your sample, what is the average age of the <var>plural( animal( 1 ) )</var>? What is the standard deviation?
    182. 

  182. 					You may round your answers to the nearest tenth.
    183. 

  183. 				</p>
    184. 

  184. 

  185. 				<div class="solution" data-type="multiple">
    186. 

  186. 					<p>
    187. 

  187. 						Average age:
    188. 

  188. 						<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>mean( DATA )</var></span> years old
    189. 

  189. 					</p>
    190. 

  190. 					<p>
    191. 

  191. 						Standard deviation:
    192. 

  192. 						<span class="sol short" data-type="decimal" data-inexact data-max-error="0.15"><var>stdDev( DATA )</var></span> years
    193. 

  193. 					</p>
    194. 

  194. 					<div class="example">decimals, like <code>0.75</code></div>
    195. 

  195. 					<div class="example">answers within <code>\pm 0.15</code> are accepted to allow for rounding part-way through</div>
    196. 

  196. 				</div> <!-- solution -->
    197. 

  197. 

  198. 				<div class="hints">
    199. 

  199. 					<p>
    200. 

  200. 						Because we only have data for a small sample of the <var>plural( POPULATION, animal( 1 ) )</var>, we are only able
    201. 

  201. 						to estimate the population mean and standard deviation by finding the <span class="hint_blue">sample mean</span>
    202. 

  202. 						<code>(\color{<var>BLUE</var>}{\overline{x}})</code> and
    203. 

  203. 						<span class="hint_pink">sample standard deviation</span> <code>(\color{<var>PINK</var>}{s})</code>.
    204. 

  204. 					</p>
    205. 

  205. 

  206. 					<div>
    207. 

  207. 						<p>
    208. 

  208. 							To find the <span class="hint_blue">sample mean</span>, add up the values of all <code class="hint_green"><var>DATA_POINTS</var></code>
    209. 

  209. 							samples and divide by <code class="hint_green"><var>DATA_POINTS</var></code>.
    210. 

  210. 						</p>
    211. 

  211. 						<p>
    212. 

  212. 							<code>
    213. 

  213. 								\color{<var>BLUE</var>}{\overline{x}} \quad = \quad
    214. 

  214. 								\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{n}} x_i}{\color{<var>GREEN</var>}{n}} \quad = \quad
    215. 

  215. 								\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} x_i}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}}
    216. 

  216. 							</code>
    217. 

  217. 						</p>
    218. 

  218. 					</div>
    219. 

  219. 

  220. 					<p>
    221. 

  221. 						<code>
    222. 

  222. 								\color{<var>BLUE</var>}{\overline{x}} \quad = \quad
    223. 

  223. 								\dfrac{<var>plus.apply( KhanUtil, DATA )</var>}{\color{<var>GREEN</var>}{<var>DATA_POINTS</var>}} \quad = \quad
    224. 

  224. 								\color{<var>BLUE</var>}{<var>MEAN</var>\text{ <var>plural( "year", MEAN )</var> old}}
    225. 

  225. 						</code>
    226. 

  226. 					</p>
    227. 

  227. 

  228. 					<p>
    229. 

  229. 						Find the <span class="hint_purple">squared deviations from the mean</span> for each sample. Since we don't know the
    230. 

  230. 						population mean, estimate the mean by using the <span class="hint_blue">sample mean</span> we just calculated
    231. 

  231. 						<code>(\color{<var>BLUE</var>}{\overline{x}} = \color{<var>BLUE</var>}{<var>MEAN</var>\text{ <var>plural( "year", MEAN )</var>}})</code>.
    232. 

  232. 					</p>
    233. 

  233. 

  234. 					<div>
    235. 

  235. 						<div class="fake_header">
    236. 

  236. 							<span style="width: 100px;">
    237. 

  237. 								Age<br/>
    238. 

  238. 								<code>x_i</code>
    239. 

  239. 							</span><span style="width: 150px;">
    240. 

  240. 								<span class="hint_gray">Distance from the mean</span>
    241. 

  241. 								<code>(x_i - \color{<var>BLUE</var>}{\overline{x}})</code>
    242. 

  242. 							</span><span style="width: 150px;">
    243. 

  243. 								<code>(x_i - \color{<var>BLUE</var>}{\overline{x}})^2</code>
    244. 

  244. 							</span>
    245. 

  245. 						</div>
    246. 

  246. 						<div class="fake_row" data-each="DATA as i, POINT">
    247. 

  247. 							<span style="width: 100px;">
    248. 

  248. 								<code><var>POINT</var></code> <var>plural( "year", POINT )</var>
    249. 

  249. 							</span><span style="width: 150px;" class="hint_gray">
    250. 

  250. 								<code><var>roundTo( 2, POINT - MEAN )</var></code> <var>plural( "year", roundTo( 2, POINT - MEAN ) )</var>
    251. 

  251. 							</span><span style="width: 150px;" class="hint_purple">
    252. 

  252. 								<code><var>SQR_DEV[ i ]</var></code> <var>plural( "year", SQR_DEV[ i ] )</var><code>^2</code>
    253. 

  253. 							</span>
    254. 

  254. 						</div>
    255. 

  255. 					</div>
    256. 

  256. 

  257. 					<div>
    258. 

  258. 						<p>
    259. 

  259. 							Normally we can find the variance <code>(\color{red}{s^2})</code> by averaging the
    260. 

  260. 							<span class="hint_purple">squared deviations from the mean</span>. But remember we don't know the real
    261. 

  261. 							population mean&mdash;we had to estimate it by using the <span class="hint_blue">sample mean</span>.
    262. 

  262. 						</p>
    263. 

  263. 						<p>
    264. 

  264. 							The age of any particular <var>animal( 1 )</var> in our sample is likely to be closer to the average age
    265. 

  265. 							of the <var>plural( DATA_POINTS, animal( 1 ) )</var> we sampled than it is to the average age
    266. 

  266. 							of all <var>plural( POPULATION, animal( 1 ) )</var> in the zoo.
    267. 

  267. 							Because of that, the <span class="hint_purple">squared deviations from the mean</span> we calculated will
    268. 

  268. 							probably underestimate the actual deviations from the population mean.
    269. 

  269. 						</p>
    270. 

  270. 						<p>
    271. 

  271. 							To compensate for this underestimation, rather than simply averaging the <span class="hint_purple">squared deviations from the mean</span>,
    272. 

  272. 							we total them and divide by <code class="hint_green">n - 1</code>.
    273. 

  273. 						</p>
    274. 

  274. 						<p>
    275. 

  275. 							<code>
    276. 

  276. 								\color{red}{s^2} \quad = \quad
    277. 

  277. 								\dfrac{\sum\limits_{i=1}^{\color{<var>GREEN</var>}{n}} (x_i - \color{<var>BLUE</var>}{\overline{x}})^2}{\color{<var>GREEN</var>}{n - 1}}
    278. 

  278. 							</code>
    279. 

  279. 						</p>
    280. 

  280. 					</div>
    281. 

  281. 

  282. 					<p>
    283. 

  283. 						<code>
    284. 

  284. 							\color{red}{s^2} \quad = \quad
    285. 

  285. 							\dfrac{<var>plus.apply( KhanUtil, jQuery.map( SQR_DEV, function( x ) { return "\\color{purple}{" + x + "}"; }) )</var>}
    286. 

  286. 							{\color{<var>GREEN</var>}{<var>DATA_POINTS</var> - 1}}
    287. 

  287. 						</code>
    288. 

  288. 					</p>
    289. 

  289. 

  290. 					<p>
    291. 

  291. 						<code>
    292. 

  292. 							\color{red}{s^2} \quad = \quad
    293. 

  293. 							\dfrac{\color{purple}{<var>roundTo( 2, sum( SQR_DEV ) )</var>}}{\color{<var>GREEN</var>}{<var>DATA_POINTS - 1</var>}} \quad = \quad
    294. 

  294. 							\color{red}{<var>VARIANCE</var>\text{ <var>plural( "year", VARIANCE )</var>}^2}
    295. 

  295. 						</code>
    296. 

  296. 					</p>
    297. 

  297. 

  298. 

  299. 					<div>
    300. 

  300. 						<p>
    301. 

  301. 							As you might guess from the notation, the sample standard deviation <code>(\color{<var>PINK</var>}{s})</code> is
    302. 

  302. 							found by taking the square root of the sample variance <code>(\color{red}{s^2})</code>.
    303. 

  303. 						</p>
    304. 

  304. 						<p>
    305. 

  305. 							<code>\color{<var>PINK</var>}{s} = \sqrt{\color{red}{s^2}}</code>
    306. 

  306. 						</p>
    307. 

  307. 					</div>
    308. 

  308. 

  309. 					<p>
    310. 

  310. 						<code>
    311. 

  311. 							\color{<var>PINK</var>}{s} = \sqrt{\color{red}{<var>VARIANCE</var>\text{ <var>plural( "year", VARIANCE )</var>}^2}} =
    312. 

  312. 							\color{<var>PINK</var>}{<var>STDDEV</var>\text{ <var>plural( "year", STDDEV )</var>}}
    313. 

  313. 						</code>
    314. 

  314. 					</p>
    315. 

  315. 

  316. 					<p><strong>
    317. 

  317. 						We can estimate that the average <var>animal( 1 )</var> at the zoo is <var>plural( MEAN, "year" )</var> old with a standard deviation
    318. 

  318. 						of <var>plural( STDDEV, "year" )</var>.
    319. 

  319. 					</strong></p>
    320. 

  320. 

  321. 				</div> <!-- hints -->
    322. 

  322. 			</div> <!-- sample -->
    323. 

  323. 

  324. 		</div> <!-- problems -->
    325. 

  325. 	</div> <!-- exercise -->
    326. 

  326. </body>
    327. 

  327. </html>
    328. 

  328.