Variance
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The document defines variance as the average of the squared differences from the mean. It provides examples of calculating variance and standard deviation for different data sets involving heights of dogs, exam scores, and word counts per page. Variance is found by taking the difference of each value from the mean, squaring it, and averaging the results. Standard deviation is the square root of the variance.
Variance
- 1. VARIANCE
- 2. WHAT IS VARIANCE ? The Variance is defined as : The average of the squared differences from the Mean.
- 3. To calculate the variance follow these steps: Work out the Mean Thenfor each number: subtract the Mean and square the result Then work out the average of those squared differences.
- 4. Example You and your friends have just measured the heights of your dogs (in millimeters):
- 5. The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm. Find out the Mean, the Variance, and the Standard Deviation. Your first step is to find the Mean: • so the mean (average) height is 394 mm. Let's plot this on the chart:
- 6. Now, we calculate each dogs difference from the Mean: To calculate the Variance, take each difference, square it, and then average the result:
- 7. QUESTIONS : 1. What is the standard deviation for the numbers: 75, 83, 96, 100, 121 and 125? 2. Ten friends scored the following marks in their end-of-year math exam: 23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82% What was the standard deviation of their marks? 3. A booklet has 12 pages with the following numbers of words: 271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314 4. What is the standard deviation number of words per page?
- 8. ANSWERS : I. 1. Firstly find the mean: Mean = (75 + 83 + 96 + 100 + 121 + 125) ÷ 6 = 600 ÷ 6 = 100 2. Next find the variance. To calculate the Variance, take each difference, square it, and then average the result: (75 - 100)2 + (83 - 100)2 + (96 - 100)2 + (100 - 100)2 + (121 - 100)2 + (125 - 100)2 = (-25)2 + (-17)2 + (-4)2 + (0)2 + (21)2 + (25)2 = 625 + 289 + 16 + 0 + 441 + 625 = 1,996 So the Variance = 1,996 ÷ 6 = 332.66... 3. The Standard Deviation is just the square root of the Variance = √(332.66...) = 18.2 correct to 1 decimal places
- 9. II. 1. Firstly find the mean: Mean = (23 + 37 + 45 + 49 + 56 + 63 + 63 + 70 + 72 + 82) ÷ 10 = 560 ÷ 10 = 56 2. Next find the variance. To calculate the Variance, take each difference, square it, and then average the result: (23 - 56)2 + (37 - 56)2 + (45 - 56)2 + (49 - 56)2 + (56 - 56)2 + (63 - 56)2 + (63 - 56)2 + (70 - 56)2 + (72 - 56)2 + (82 - 56)2 = (-33)2 + (-19)2 + (-11)2 + (-7)2 + (0)2 + (7)2 + (7)2 + (14)2 + (16)2 + (26)2 = 1,089 + 361 + 121 + 49 + 0 + 49 + 49 + 196 + 256 + 676 = 2,846 So the Variance = 2,846 ÷ 10 = 284.6 3. The Standard Deviation is just the square root of the Variance = √(284.6) = 16.9 correct to 1 decimal place
- 10. III. Firstly find the mean number of words per page: Mean = (271 + 354 + 296 + 301 + 333 + 326 + 285 + 298 + 327, + 316 + 287 + 314) ÷ 12 =3,708 ÷ 12 = 309 2. Next find the variance. To calculate the Variance, take each difference, square it, and then average the result: (271 - 309)2 + (354 - 309)2 + (296 - 309)2 + (301 - 309)2 + (333 - 309)2 + (326 - 309)2 + (285 - 309)2 + (298 - 309)2 + (327 - 309)2 + (316 - 309)2 + (287 - 309)2 + (314 - 309)2 = (-38)2 + (45)2 + (-13)2 + (-8)2 + (24)2 + (17)2 + (-24)2 + (- 11)2 + (18)2 + (7)2 + (-22)2+ (5)2 = 1,444 + 2,025 + 169 + 64 + 576 + 289 + 576 + 121 + 324 + 49 + 484 + 25 = 6,146 So the Variance = 6,146 ÷ 12 = 512.166... 3. The Standard Deviation is just the square root of the Variance = √(512.166...) = 22.6 correct to 1 decimal place
- 11. Questions 1.What is the standard deviation for the numbers: 75, 83, 96, 100, 121 and 125? 2.A booklet has 12 pages with the following numbers of words: 271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314.Find its variance. 3.Nine friends guessed the number of marbles in a jar. When the answer was revealed they found they had guessed well (and one was the winner!) Here is how close they each got: -9, -7, -4, -1, 0, 2, 7, 9, 12. Find the variance.