# The standard deviation in context · Grade 11 · 3 units > Explanation, worked examples and free practice following the Israeli curriculum. In The standard deviation in context (Grade 11 · 3 units) you practise 5 skills: Which group is more spread out · The standard deviation of a list · From a table or a chart · Centre and spread of two groups · What happens to the standard deviation. Two groups of data can have the same mean and still look very different. In one group the values are close together, and in the other they are spread out. A large spread shows that a group is varied, and a small spread shows that it is uniform. The standard deviation measures the spread about the mean. Work out the mean, the distance of each value from it, and the square of each distance. Work out the mean of the squares; the standard deviation is its square root. Divide by the number of all the values; the standard deviation is in the same units as the data. In a frequency table, a bar chart or a pie chart, each value counts as many times as it appears: multiply its squared distance by its frequency. A spreadsheet or a calculator works out the standard deviation quickly, but it helps to know what it works out. When the same number is added to every value, the standard deviation does not change. When every value is multiplied by a number, it is multiplied by that number too. Adding a value equal to the mean makes the standard deviation smaller. A high mean does not mean a high standard deviation: describe a group by its centre and by its spread. **Tip:** Before working anything out, look at the values: if they are far from the mean, the standard deviation is large. ## What you learn ### Which group is more spread out - Example: Here are some pupils’ marks in a test, in two groups. Group A: 79, 79, 79, 79 and 64. Group B: 58, 82, 94, 76 and 70. In which group are the values more spread out? - Options: Group A · Group B - Answer: Group B - Solution: In both groups the mean is 76. In Group B the values are further from the mean. The standard deviation in Group A is 6, and in Group B it is 12. So the values in Group B are more spread out. ### The standard deviation of a list - Example: Here are some pupils’ marks in a test: 80, 80, 82, 88, 76 and 86. What is the standard deviation of all the values? - Answer: 4 - Solution: The mean: 492 ÷ 6 = 82 The squared distances from the mean are 4, 4, 0, 36, 36 and 16, and their mean is: 96 ÷ 6 = 16 The standard deviation is the square root of that number: √16 = 4 ### From a table or a chart - Example: The table shows how many books each pupil in a class read in one month. (A frequency table of how many books each pupil in a class read in one month) What is the median of the values? - Answer: 3 - Solution: There are 24 values, so the median is the mean of the values in places 12 and 13 in order: (3 + 3) ÷ 2 = 3 ### Centre and spread of two groups - Example: Here are some pupils’ marks in a test, in two groups. Group A: 75, 87, 84, 78, 69 and 75. Group B: 71, 50, 59, 77, 71 and 62. The mean in Group A is 78, and in Group B it is 65. Which group has the larger standard deviation? - Options: Group A · Group B - Answer: Group B - Solution: In Group A, the squared distances from the mean are 9, 81, 36, 0, 81 and 9, and their mean is: 216 ÷ 6 = 36 The standard deviation is the square root of that number: √36 = 6 In Group B, the squared distances from the mean are 36, 225, 36, 144, 36 and 9, and their mean is: 486 ÷ 6 = 81 The standard deviation is the square root of that number: √81 = 9 So the standard deviation in Group B is larger. ### What happens to the standard deviation - Example: Here are some families’ monthly electricity bills, in shekels: 460, 460, 480 and 400. One more value is added, and the mean stays the same. Which value was added? - Answer: 450 - Solution: The mean: 1800 ÷ 4 = 450 The value added is equal to the mean, that is 450. ## Not on the site yet - The standard deviation in a spreadsheet - Explaining in words why the standard deviation did not change The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 3 units (grades 10 and 11). [Practise on the site](https://kixplo.com/il/en/) · [All Grade 11 · 3 units topics](https://kixplo.com/il/en/math/grade-11-3u/)