aStandard Deviation: ≈ 2 home runs Interpretation: The player's typical home runs in their rookie season differ from the mean by about 2 home runs.
B
bStandard Deviation: ≈ 3 home runs Interpretation: The player's typical home runs for this season differ from the mean by about 3 home runs.
C
cSample Answer: This season's home runs are more spread out. The player's performance is better this season.
Practice makes perfect
a Consider the data for the player's rookie season.
1, 0, 6, 2, 0, 3
We are asked to find the standard deviation for the given data. To do so, let's first find the mean of the data.
Mean
The mean of a numerical data set is the sum of the data divided by the number of data values. Since we have 6 data values, the sum of the data values will be the sum of the data values divided by 6.
The standard deviation σ 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 out the mean of the data set.
x=12/6 ⇒ x=2
With this value we can now calculate the deviation and the square of each deviation. Let's do this in a table.
x
x
x-x
(x-x)^2
1
2
-1
1
0
2
-2
4
6
2
4
16
2
2
0
0
0
2
-2
4
3
2
1
1
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(4.3) ⇒ σ≈ 2
Therefore, the standard deviation is about 2. This means that the player's typical home runs in Rookie season differ from the mean by about 2 home runs.
b Consider the data for this season.
4,6,4,8,7,13
Again, we are asked to calculate the standard deviation for the given data. Proceeding in a similar way as we did in Part A, let's first calculate the mean and then the standard deviation.
Mean
In this case, we also have 6 data values. Therefore, the mean will be the sum of the data divided by 6.
Finally, we will take the square root of the variance to get the standard deviation.
σ=sqrt(9.3) ⇒ σ≈ 3
Therefore, the standard deviation is about 3. This means that the player's typical home runs this season differ from the mean by about 3 home runs.
c We are asked to compare the standard deviation of the rookie season and this season. Let's first recall the standard deviation of each season.
& Standard Deviation
Rookie Season:& 2 home runs
This Season:& 3 home runs
Since this season's standard deviation is greater, home runs this season are more spread out. Additionally, the player's performance is better this season, since they made at least 4 home runs for month. Please note that there are many possible interpretations. Here we are only showing one possibility.
