Consider some arbitrarydata set. Calculate each measure before and after dividing the same amount from each data value.
See solution.
Practice makes perfect
We want to determine how dividing each value in a data set by the same nonzero number affects a variety of measures — mean, median, mode, and range. We will analyze each measure one at a time.
Mean
Let's 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
Recall that the mean is equal to the sum of the data values S divided by total number of data values n.
Mean = S/n
Let r denote the nonzero amount that we will use to divide each data value. After we divide each data value by r, the sum of all of the data values will be also divided by r.
a_1 + ... + a_n_(nterms) &= S [1.5em]
a_1/r + ... + a_n/r_(nterms) &= S/r
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 just need to pick the value that is in the middle. Let m denote the middle value. We will also divide each data value by r.
a_1, ... m, ... a_n
↓
a_1/r, ... m/r, ... a_n/r
When we divide each data value by r, the data values will still remain ordered. Additionally, the middle value is in exactly the same position. Therefore, we obtain the median of the modified data set by dividing the median of the initial data set by r.
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 divide each data value by r.
a_1/r, ... m_1/r, m_2/r, ... a_n/r
Let's calculate the median of the modified data set.
We can see that, once again, the median of the modified data set is equal to the median of the initial data set divided r.
Mode
The mode of the 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 divide each data value by r, 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 divide each data value by r, the same values will appear the same number of times. However, all of the values will be divided by r. Therefore, the mode will also be divided by r.
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 divide each data value by r, including g and l. For simplicity, we will assume that r is positive. Then the signs of the data values will not change and the data values will remain in the same order.
l, a_2, ... a_(n-1), g
↓
l/r, a_2/r, ... a_(n-1)/r, g/r
We will again calculate the range, but this time we will use the modified data values.
Notice that the range equals the range of the initial data set divided by r. Therefore, the subtraction does not affect the range.
Conclusions
Finally, we can summarize how the division affects the various measures. To obtain the measures of the modified data set, we can divide the measures of the initial data set by the same amount.
