We want to describe the effect that adding 10 to every value in a will have on the mean, median, mode, range, and box-and-whisker plot. Then we also want to consider the case of multiplying every value by 10. Let's start with adding the !
Adding the Constant to Every Value
Let's analyze what happens with the listed statistics after adding 10 to every value, one at a time.
Mean
Consider an example data set.
{ x_1, x_2, ..., x_(n-1), x_n }
Let x represent the of our example data set.
x=x_1+x_2+...+x_n/n
Now, let's add 10 to each value x to obtain the new mean x'.
x'=(x_1+ 10)+(x_2+ 10)+...+(x_n+ 10)/n
Since we have n values, we added the constant n times. Let's rearrange the terms in the numerator using the .
x'=x_1+x_2+...+x_n+10+10+...+10^n/n ⇕ x'=x_1+x_2+...+ x_n+10 n/n
Notice that we can rewrite the fraction as a sum of two fractions.
x'=x_1+x_2+...+ x_n+10n/n
x'=x_1+x_2+...+ x_n/n+10n/n
x'=x_1+x_2+...+ x_n/n+10
x'= x+10
We can see that the new mean is 10 more than the initial mean.
Median
The is the middle value in a data set. Let's assume that our example data set is ordered and n is an . We can call this middle value x_\text{M}.
{ x_1, x_2, ..., x_M^(Median), ..., x_(n-1), x_n }
Adding the constant to every value does not change the order of the data values. This means that the new middle value will be the initial middle value increased by 10.
{ x_1+ 10,..., x_M+ 10^(Median), ..., x_n+ 10 }
After adding the constant to every data value, the new median is 10 more than the initial one. This statement will also hold true also for any n that is an .
Mode
The is the most frequently occurring value in a data set. Let's consider an example data set in which x represents the mode.
{a, x, x, x, x, ..., y, z}
Since we are adding the exact same number to each data value, the new mode will be also increased by this number.
{a+ 10,x+ 10, x+ 10, x+ 10, x+ 10, ..., z+ 10}
The new mode after adding 10 to every data value is the initial mode increased by 10.
Range
Now let's think about the , which is the difference between the maximum and minimum values in a data set. Again let's take a look at our example data set. We will assume that this data set is ordered from least to greatest.
{ x_1 , x_2, ..., x_(n-1),x_n }
We can see that the range of this data set is x_n-x_1. Next, let's see what happens to the range after adding 10 to each data value.
{ x_1+ 10 , x_2+ 10, ..., x_(n-1)+ 10,x_n+ 10 }
The maximum value of the new data set is x_n+10 and the minimum value is x_1+10. Let's evaluate the difference of these values!
Range=Maximum Value - Minimum Value
Range= x_n+10-( x_1+10)
Range=x_n+10-x_1-10
Range=x_n-x_1+10-10
Range=x_n-x_1
We can see that adding 10 to every value in a data set does not affect the range.
Box-and-Whisker Plot
Finally we can use our previous results to analyze the effect that adding 10 to every data value will have on the . First let's recall that we use to create these graphs — the maximum, the minimum, the median, and the first and third .
We found that, after adding 10 to every value, the minimum and maximum values, as well as the median, are increased by 10. Following the same logic, since the quartiles, Q_1 and Q_3, are the medians of the lower and upper halves, they will also be increased by 10. Therefore, the new graph will be shifted 10 units right.
Notice that the distance between the whiskers stays the same because adding the constant did not affect the range.
Multiplying Every Value by the Constant
This time we want to identify the effect of multiplying each value in a data set by 10. As we did before, we will start by analyzing what happens to the mean when we multiply each value by the same number.
Mean
We will use the same example data set with the initial mean x.
x=x_1+x_2+...+x_n/n
Let's multiply each value x by 10 to obtain the new mean x^*.
x^*=10x_1+ 10x_2+...+ 10x_n/n
Now, we can factor out 10 from the expression in the numerator.
x^*=10x_1+10x_2+...+10x_n/n
x^*=10(x_1+x_2+...+x_n)/n
x^*=10*x_1+x_2+...+x_n/n
x^*=10 x
We can see that the new mean is 10 times greater than the initial mean.
Median
Again, we will assume that our example data set is ordered and n is odd. Yet again, we will let x_\text{M} be our middle value.
{ x_1, x_2, ..., x_M^(Median), ..., x_(n-1), x_n }
Multiplying every value by a constant also does not change the order of the data values. Therefore, the new middle value will be the initial middle value multiplied by 10.
{ 10x_1, 10x_2,..., 10x_M^(Median), ..., 10x_(n-1), 10x_n }
After multiplying every data value by the constant, the new median is 10 times more than the initial one. Notice that this statement is also true also for even numbered values of n.
Mode
Let's consider an example data set in which x represents the mode, as we did in the case of addition.
{a, x, x, x, x, ..., y, z}
Since we are multiplying each data value by the exact same number, the new mode will be also multiplied by this number.
{ 10a, 10x, 10x, 10x, 10x, ..., 10z}
The new mode after multiplying every data value by 10 is the initial model increased by a factor of 10.
Range
For the range, let's take a look at the example data set and this time we will again assume that this data set is ordered from least to greatest.
{ x_1 , x_2, ..., x_(n-1),x_n }
The range of this data set is x_n-x_1. Now, let's multiply each data value by 10.
{ 10 x_1 , 10x_2, ..., 10x_(n-1), 10x_n }
We can see that the maximum value of the new data set is 10x_n and the minimum value is 10x_1. Let's evaluate the difference of these values!
Range=Maximum Value - Minimum Value
Range= 10x_n- 10x_1
Range=10(x_n-x_1)
In the case of multiplying by a constant, the range is affected. After multiplying every data value by 10, the range increased by 10 times.
Box-and-Whisker Plot
As before, we will use our previous results to analyze the effect that multiplying every data value by 10 will have on the box-and-whisker plot. We know that, after multiplying every value by 10, the minimum value, maximum value, and median are all increased by a factor of 10. Using the same reasoning, the quartiles will also be increased by a factor of 10.
The new graph is shifted to the right but not by a constant value as before. Also, the distance between the whiskers is 10 times greater because multiplying by a constant affects the range. This applies to other distances between the corresponding values as well.
Conclusions
Finally, we can gather all of our results and present them using a table.
|
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Adding the Constant
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Multiplying by the Constant
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| Mean
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Increased by 10
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Increased 10 times
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| Median
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Increased by 10
|
Increased 10 times
|
| Mode
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Increased by 10
|
Increased 10 times
|
| Range
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No effect
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Increased 10 times
|
| Box-and-Whisker Plot
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Shifted 10 units right
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Shifted and stretched by a factor of 10
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These results can be applied accordingly in any case of adding a constant to every data value or multiplying every data value by a constant, no matter what that constant is.
