If every value in the data set is increased by a constant k, the mean, median, and mode of the new data set can be found by adding k to each of the original statistics. The range and the standard deviation will not change.
Mean: 3.35 Median: 2.95 Mode: Does not exist Range: 3.3 Standard Deviation: Approximately 1.09
Practice makes perfect
Let's consider the given data set and change mixed numbers into decimal numbers.
Altitudes of the Airplanes [0.8em]
1 910mi, 1 25mi,
4 15mi, 1 12mi,
1 15mi, 910mi
⇓
1.9, 1.4, 4.2, 1.5, 1.2, 0.9
We want to find the mean, median, mode, range, and standard deviation of the data set obtained by adding the given constant, k= 1.5, to each value.
If every value in the data set is increased by the constant 1.5, then the statistics of the new data set will behave in a consistent, predictable way.
The new mean, median, and mode can be found by adding 1.5 to the mean, median, and mode of the original data set.
The range and standard deviation will not change.
Notice that only the measures of center are increased by the constant and the measures of spread will not change. This is because the distances between the individual values do not change.
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 6 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.
0.9 , 1.2 , 1.4 | 1.5 , 1.9 , 4.2
Since there are 6 values, there is no one middle value. Therefore, the median is the mean of the two middle values.
Median: 1.4+ 1.5/2=1.45
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.
1.9 , 1.4 , 4.2 , 1.5 , 1.2 , 0.9
Since the data set does not contain any repeated values, there is no mode.
Range
The range is the difference between the least and greatest values in a set of data.
1.9 , 1.4 , 4.2 , 1.5 , 1.2 , 0.9
For this set, the greatest value is 4.2 and the least value is 0.9.
Range: 4.2- 0.9=3.3
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.
Standard Deviation
sqrt((x_1-x )^2 + (x_2-x )^2 + ... + (x_n-x )^2/n)
In this formula, x_1, ... ,x_n are the values of the set of data, x is the mean, and n is the number of values. We have 6 values and the mean is x=1.85. Let's apply the formula to each value in the set.
x_n
x_n-x
(x_n-x)^2
1.9
1.9 - 1.85=0.05
0.05^2=0.0025
1.4
1.4 - 1.85=- 0.45
(- 0.45)^2=0.2025
4.2
4.2 - 1.85=2.35
2.35^2=5.5225
1.5
1.5 - 1.85=- 0.35
(- 0.35)^2=0.1225
1.2
1.2 - 1.85=- 0.65
(- 0.65)^2=0.4225
0.9
0.9 - 1.85=- 0.95
(- 0.95)^2=0.9025
Sum of Values
7.175
Finally, since n= 6, we need to divide by 6 and then calculate the square root.
Standard Deviation: sqrt(7.175/6)≈ 1.09
Adding a Constant
Finally, we can find new values of the statistics by adding 1.5 to the mean, median, and mode.
