Consider some arbitrarydata set. Calculate each measure before and after subtraction.
See solution.
Practice makes perfect
We want to determine how subtracting the same number from each value in a data set affects a variety of measures — mean, median, mode, and range. We will analyze each measure one at a time.
Mean
Let's first consider some arbitrary data set with n data values. For simplicity of the notation, we will assume that the data values are arranged in increasing order.
a_1, a_2, ... a_(n-1), a_n
Recall that the mean is equal to the sum of the data values S divided by the total number of data values n.
Mean = S/n
Let d denote the amount that we subtract from each data value. After we subtract d from each data value, the sum of all of the data values will be n times d less than the initial sum.
a_1 + ... + a_n_(nterms) &= S [1.7em]
(a_1- d) + ... + (a_n- d)_(nterms) &= S - n d
We can now calculate the mean of the modified data set.
Recall that the data values are arranged in order. When we have an odd number of data values, we need to choose the value that is in the middle. Let m denote the middle value of our data set. We will subtract d from each data value.
a_1, ... m, ... a_n
↓
a_1 - d, ... m - d, ... a_n - d
When we subtract d from each data value, the order of the elements does not change. Additionally, the middle value is in exactly the same position. Therefore, the median decreases by d.
Case 2
This time, we need to take into account two middle values and calculate their mean. Let m_1 and m_2 be the middle data values.
a_1, ... m_1, m_2, ... a_n
We can find the median by calculating the mean of m_1 and m_2.
Median = m_1+ m_2/2
Similar to Case 1, we will now subtract d from each data value.
a_1 - d, ... m_1 - d, m_2 - d, ... a_n - d
Let's calculate the median of the modified data set.
We can see that, once again, the median decreased by d.
Mode
The mode of a data set is the data item or items that occurs the most often. Recall that we can have no mode, one mode, or more than one mode.
Number of Modes
Definition
How Subtracting d Will Affect The Mode
0
There are no values that appear more than the other values in the data set.
When we subtract d from each data value, the frequency of the values will not change and there will be still no mode.
1
There is one value that appears more often than any other value in the data set.
When we subtract d from each data value, the same values will appear the same number of times. However, all of the values will decrease by d. Therefore, the mode will also decrease by d.
2+
There are two or more values that appear an equal number of times and they appear more often than other data values.
Range
We will now determine how modifying the data values affects the range of the data set. Let g and l be the greatest and the least data value, respectively.
l, a_2, ... a_(n-1), g
Recall that the range is the difference between the greatest and the least data value.
Range = g - l
Let's now subtract d from each of the data values, including g and l.
l, a_2, ... a_(n-1), g
↓
l - d, a_2 - d, ... a_(n-1) - d, g - d
We will again calculate the range, but this time we will use the modified data values.
