In order to compare the two data sets, we need to calculate the various measures of spread and center for each type of singer. Let's take a look at these one at a time!
Finding the Mean
The is the average of a set of data. We need to find the sum of the ages of each type of singer and then divide by the number of singers in that data set. Let's start with the sopranos.
Mean=Sum of values/Number of values
63+42+28+45+36+48+32+40+57+49/10
440/10
44
The mean of the soprano singers' age is 44. Now, we can calculate the mean of the bass singers in the same way.
Mean=Sum of values/Number of values
32+34+53+35+43+41+29+35+24+34/10
360/10
36
The mean of the bass singers' age is 36. This tells us that the average age of the bass singers is younger than that of the sopranos.
Finding the Median
The is the middle value, or the average of the two middle values, when the data is listed from least to greatest. First, we will need to put the data in numerical order. Let's start with the sopranos.
28, 32, 36, 40, | 42, 45, | 48, 49, 57, 63
Now, we can find the average of the two values in the middle.
42+45/2 ⇒ 43.5
The median age of the soprano singers is 43.5 We will follow the same process to find the median of the bass singers. Let's start by putting them into numerical order.
24, 29, 32, 34, | 34, 35, | 35, 41, 43, 53
Once again, we can find the average of the two values in the middle.
34+35/2 ⇒ 34.5
The median age of the bass singers is 34.5. Similar to the mean, the bass singers have a younger median age than the sopranos.
Finding the Range
The is the difference between the greatest and least values. Let's look at the sopranos' ages in numerical order.
28, 32, 36, 40, 42, 45, 48, 49, 57, 63
When we take the difference of the extreme values, 63-28, we find that the range of the soprano singers ages is 35 years. Let's do the same thing with the bass singers now.
24, 29, 32, 34, 34, 35, 35, 41, 43, 53
When we find the difference of the extremes of this set, 53-24, we find that the range of the bass singers ages is 29 years. This tells us that the bass singers are closer in age to each other than the soprano singers are.
Finding the Interquartile Range
To find the interquartile range, we need to find the median of the lower half, the first quartile, and the median of the upper half, the third quartile. Let's start with the soprano singers.
Median_L Median_U
↓ ↓
28, 32, 36, 40, 42, 45, 48, 49, 57, 63
Lower Half Upper Half
The first quartile is 36 and the third quartile is 49. The difference between the two values is the interquartile range.
49-36=13
The soprano singers ages has an interquartile range of 13. We can now do the same process with the bass singers' ages.
Median_L Median_U
↓ ↓
24, 29, 32, 34, 34, 35, 35, 41, 43, 53
Lower Half Upper Half
The first quartile is 33 and the third quartile is 41. Taking the difference will, once again, find the interquartile range.
41-33=8
The bass singers ages has an interquartile range of 8. This, just as the range did, shows that the bass singers ages are closer together than the soprano singers.
Finding the Standard Deviation
There are four steps to finding the of a set of data.
- Find the mean of the data.
- Find the squared deviation of each data value.
- Find the mean of the squared deviations.
- Find the square root of the mean of the squared deviations.
Luckily, we already know the means.
- Mean of soprano singers: 44
- Mean of bass singers: 36
Now, we can find the deviation from the mean for each soprano's age and square it. Let's use a table to organize the information.
| Soprano Age
|
Deviation from Mean
|
Squared Deviation
|
| 63
|
63-44=19
|
19^2 = 361
|
| 42
|
31-44=5
|
5^2=25
|
| 28
|
28-44=-16
|
(-16)^2=256
|
| 45
|
45-44=1
|
1^2 = 1
|
| 36
|
36-44=-8
|
(-8)^2=64
|
| 48
|
48-44=4
|
4^2=16
|
| 32
|
32-44=-12
|
(-12)^2=144
|
| 40
|
40-44=-4
|
(-4)^2=16
|
| 57
|
57-44=13
|
13^2=169
|
| 49
|
49-44=5
|
5^2=25
|
Now we can find the mean of the squared deviations.
Mean=Sum of values/Number of values
361+25+256+1+64+16+144+16+169+25/10
1077/10
107.7
The final step is to find the square root of the mean of the square deviations.
sqrt(107.7) ≈ 10.4
The standard deviation of the soprano singer is approximately 10.4. Let's go through the same process for the bass singers.
| Bass Age
|
Deviation from Mean
|
Squared Deviation
|
| 32
|
32-36=-4
|
(-4)^2 = 16
|
| 34
|
34-36=-2
|
(-2)^2=4
|
| 53
|
53-36=17
|
17^2=289
|
| 35
|
35-36=-1
|
(-1)^2 = 1
|
| 43
|
43-36=7
|
7^2=49
|
| 41
|
41-36=5
|
5^2=25
|
| 29
|
29-36=-7
|
(-7)^2=49
|
| 35
|
35-36=-1
|
(-1)^2=1
|
| 24
|
24-36=-12
|
(-12)^2=144
|
| 34
|
34-36=-2
|
(-2)^2=4
|
Next, we can find the mean of the squared deviations.
Mean=Sum of values/Number of values
16+4+289+1+49+25+49+1+144+4/10
582/10
58.2
One more time, the final step is to find the square root of the mean of the square deviations.
sqrt(58.2) ≈ 7.6
The standard deviation of the bass singers is approximately 7.6. Since the standard deviation for the bass singer is less than the value for the sopranos, this tells us that the ages for the bass singers do not vary as much as the ages of the soprano singers.
