We want to find the mean, median, mode, range, and standard deviation of data set obtained by multiplying the given constant, k=0.5, by each value.
If every value in the data set is multiplied by the constant 0.5, the mean, median, mode, range, and standard deviation of the new data set can be found by multiplying each original statistic by 0.5.
To begin, let's find the statistics of the original data set.
Mean
The mean of a data set x is calculated by finding the sum of all of the values in the set and then dividing by the number of values in the set. In this case, there are 12 values in the set.
When the data are arranged in numerical order, the median is the middle value — or the mean of the two middle values. Let's arrange the given values and find the median.
36,37,37,38,39,41∣42,42,42,44,45,49
Since there are 12 values, there is no one middle value. Therefore, the median is the mean of the two middle values.
Median: 241+42=41.5
Mode
The mode is the value or values that appear most often in a set of data.
Let's find the mode of the given values.
42,39,45,44,37,42,38,37,41,49,42,36
The value that appears most often is 42, so this is our mode.
Range
The range is the difference between the least and greatest values in a set of data.
42,39,45,44,37,42,38,37,41,49,42,36
For this set, the greatest value is 49 and the least value is 36.
Range:49−36=13
Standard Deviation
The standard deviation of a set of data is the average amount by which each individual value deviates or differs from the mean.
In this formula, x1,…,xn are the values of the set of data, x is the mean, and n is the number of values. We have 12 values and the mean is x=41. Let's apply the formula to each value in the set.
x
x−x
(x−x)2
42
42−41=1
12=1
39
39−41=-2
(-2)2=4
45
45−41=4
42=16
44
44−41=3
32=9
37
37−41=-4
(-4)2=16
42
42−41=1
12=1
38
38−41=-3
(-3)2=9
37
37−41=-4
(-4)2=16
41
41−41=0
02=0
49
49−41=8
82=64
42
42−41=1
12=1
36
36−41=-5
(-5)2=25
Sum of Values
=162
Finally, since n=12, we need to divide by 12 and then calculate the square root.
StandardDeviation:12162≈3.67
Multiplying by a Constant
Finally, we can find new values of the statistics by multiplying each value by 0.5.
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