a The mean is the mathematical average. The median is the middle value. The mode is the most repeated value. The range is the difference between the greatest and least value.
bSample Answer: The new student increases all the measures except the mode. Explanation: See solution.
Practice makes perfect
a Consider the given data heights in inches of our classmates. Please note that there are infinitely possible data sets. Here we are only showing one possibility.
Height (inches)
69
60
65
68
67
67
63
62
64
61
We are asked to find the measures of center and variability of the heights. We will calculate each of them one at a time.
Mean
The mean of a numerical data set is the sum of the data divided by the number of data values. Since we have 10 values, we will divide the sum of the data by 10.
The median of a numerical data set is the middle number when the values are written in numerical order. If the data set has an even number of values, the mean of the two middle values will be the median. Let's write the data in numerical order.
60, 61, 62, 63, 64, 65, 67, 67, 68, 69
Since we have an even number of data values, the mean of the two middle values will be the median.
Median:64+65/2=64.5inches
Mode
The mode of a data set is the value or values that occur most often. Let's take a loot at our data.
60, 61, 62, 63, 64, 65, 67, 67, 68, 69
We can see that 67 is the value that occurs most often. Therefore, 67 inches is the mode.
Range
The range of a data set is the difference of the greatest value and the least value. In this case, the greatest value is 69 and the least value is 60.
Range:69-60=9inches
Standard Deviation
The standard deviation of a numerical data set is given by the following formula.
In this formula, n is the number of data values in the data set, x_1, x_2,..., x_n are the data values, and x_1-x, x_2-x,... x_n-x are the deviations of each data value. The deviation is given by the difference of the data value and the mean of the data set. We already found the mean of the data set.
x=646/10 ⇒ x=64.6inches
With this value we can calculate the deviation and the square of each deviation. Let's do this in a table.
x
x
x-x
(x-x)^2
69
64.6
4.4
19.36
65
64.6
0.4
0.16
67
64.6
2.4
5.76
63
64.6
-1.6
2.56
64
64.6
-0.6
0.36
60
64.6
-4.6
21.16
68
64.6
3.4
11.56
67
64.6
2.4
5.76
62
64.6
-2.6
6.76
61
64.6
-3.6
12.96
Now, let's find the mean of the squared deviations. This is called the variance. Let's do it!
Finally, we will take the square root of the variance to get the standard deviation.
σ=sqrt(8.64) ⇒ σ≈ 3
Therefore, the standard deviation is about 3 inches. This means that a student's typical height differs from the mean by about 3 inches.
b Given that a new student who is 7 feet tall joins our class, we are asked how this value affects the measures we found in Part A. Let's first convert the given height into inches.
7 feet = 84 inches
We can see that this value is much greater than our original data. Therefore, the given height is an outlier. This value will increase all the measures we found in Part A except the mode. To verify this, let's calculate the measures after the new value is added.
Mean
In this case, the mean will be the sum of the values divided by 11.
Therefore, the mean with the outlier is about 66.4 inches. It is greater than the original mean 64.6. Let's calculate the difference of these means.
66.4-64.6=1.8inches
Therefore, the new value increases the mean by about 1.8 inches.
Median
Let's write the given data in numerical order to find the median.
60, 61, 62, 63, 64, 65, 67, 67, 68, 69, 84
We can see that the median is now 65 inches. Comparing it with the original median, 64.5, we can see that the new median is greater. Let's find the difference of these medians.
65-64.5=0.5inches
Therefore, the new value increases the median by 0.5 inches.
Mode
Again, let's take a look at the values to find the mode.
60, 61,62, 63, 64, 65, 67, 67, 68, 69, 84
In this case, the mode is not affected. It is still 67 inches.
Range
For the range, we can see that the greatest value is now 84 and the least value is still 60.
Range: 84-60=24 inches
Therefore, the range is now 24 inches. It is greater than the original range 9. Let's calculate the difference of the ranges.
24-9=15inches
The new value increases the range by 15 inches.
Standard Deviation
Now, by using the mean x=66.4 we can calculate the deviation and the square of the each deviation. Let's do this in a table.
Finally, we will take the square root of the variance to get the standard deviation.
σ=sqrt(38.96) ⇒ σ≈ 6.2
The standard deviation is about 6 inches. It is greater than the original standard deviation 3. Therefore, the new value increases the standard deviation by about 3 inches.
