Answer :
Final answer:
The standard deviation increases proportionally when each data point in a set is multiplied by a constant. This is demonstrated by a data set (17, 11, 15, 14, 16) that when multiplied by a constant (4), results in the standard deviation being four times larger. Additionally, it's unnecessary to redo all calculations when the measurements change, for example from miles to kilometers, the standard deviation can just be multiplied by the conversion factor.
Explanation:
When you are asked to compute the standard deviation 's' for the data points 17, 11, 15, 14, 16, you will do so by following the formula for calculating standard deviation, which includes summing the squares of the differences between each data point and the mean, before dividing by the number of data points, and taking the square root. This will result in a value of 's' which we rounded to 1 decimal place, as per your instructions.
For part 'b', when each data point is multiplied by the constant 4, the resulting data set 68, 44, 60, 56, 64 will give a computed standard deviation 's' of 4. The new 's' is four times as large as the initial 's', corresponding with the constant we multiplied by.
The standard deviation increases by a factor equal to the absolute value of the constant when each data point in a data set is multiplied by a constant 'c'. This leads us to believe that the standard deviation does not remain the same, nor does it become smaller, but is |c| times larger when multiplying each data observation by c.
Finally, you do not need to redo all the calculations if you change to measurements in kilometers. Since 1 mile is approximately 1.6 kilometers, simply multiply your current standard deviation by 1.6 to obtain your new 's' in kilometers. Thus, the standard deviation in kilometers would be approximately 3.36 (rounded to two decimal places, as indicated).
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Answer:
Step-by-step explanation:
a) Mean = (17 + 11 + 15 + 14 + 16)/5 = 14.6
Standard deviation = √(summation(x - mean)²/n
n = 5
Summation(x - mean)² = (17 - 14.6)^2 + (11 - 14.6)^2 + (15 - 14.6)^2 + (14 - 14.6)^2 + (16 - 14.6)^2 = 21.2
Standard deviation = √(21.2/5 = 2.06
Approximating to 1 decimal place, s = 2
The new data set is
68, 44, 60, 56, 64
Mean = (68 + 44 + 60 + 56 + 64)/5 = 58.4
Summation(x - mean)² = (68 - 58.4)^2 + (44 - 58.4)^2 + (60 - 58.4)^2 + (56 - 58.4)^2 + (64 - 58.4)^2 = 339.2
Standard deviation = √(339.2/5 = 8.24
c) The standard deviation of the new data is 4 times the standard deviation of the previous data
In general, multiplying each data value by the same constant c results in the standard deviation being |c| times as large.
d) s = 2.1 miles
Since 1 mile = 1.6 kilometers, the constant with which we would multiply the given standard deviation is 1.6. Therefore, converting to kilometers, it becomes
1.6 × 2.1 = 3.36 km
