A distribution has a mean of 300 and a standard deviation of 25. If the z score of quartile 1 is -.7 and the a score of Q3 is 0.7, what values would be considered outliers?
Here is my work so far
-.7 = Q1-300/25 Q1 = 282.5
.7 = Q3 -300/25 Q3 =317.5
IQR = 35
Q1 -52.5 =230
Q3 +52.5 =370
No lower bound outliers, but is there outliers above upper bound?
Well, well, well, outliers, huh? They're like the unexpected guests of the statistical world! Alright, let's have a chuckle and calculate those puppies.
To find the values considered outliers, we need to first convert those sassy z-scores into their corresponding raw scores. Now, the formula for that is raw score = (z-score * standard deviation) + mean.
For Q1, we have a z-score of -0.7. Plugging that into the formula, we get:
raw score = (-0.7 * 25) + 300
raw score = 282.5
And for Q3, we have a z-score of 0.7. Plugging it in, we get:
raw score = (0.7 * 25) + 300
raw score = 317.5
So, any value that falls below 282.5 or above 317.5 would be considered an outlier. They're the little rebels of the data set, marching to the beat of their own drum!
To find the values considered outliers in a distribution, we need to determine the lower and upper bounds for outliers based on the quartiles and z scores.
Given that the z score of Quartile 1 (Q1) is -0.7 and the z score of Quartile 3 (Q3) is 0.7, we can use the formula:
z = (x - mean) / standard deviation
First, let's find the z-score for Q1:
z(Q1) = -0.7
-0.7 = (Q1 - 300) / 25
Rearranging the equation to solve for Q1:
Q1 - 300 = -0.7 * 25
Q1 - 300 = -17.5
Q1 = 300 - 17.5
Q1 = 282.5
So, Quartile 1 (Q1) is equal to 282.5.
Next, let's find the z-score for Q3:
z(Q3) = 0.7
0.7 = (Q3 - 300) / 25
Rearranging the equation to solve for Q3:
Q3 - 300 = 0.7 * 25
Q3 - 300 = 17.5
Q3 = 300 + 17.5
Q3 = 317.5
So, Quartile 3 (Q3) is equal to 317.5.
To determine the outliers, we can calculate the lower and upper bounds using the interquartile range (IQR).
IQR = Q3 - Q1 = 317.5 - 282.5 = 35
The lower bound for outliers is calculated as Q1 - (1.5 * IQR):
Lower bound = 282.5 - (1.5 * 35) = 282.5 - 52.5 = 230
The upper bound for outliers is calculated as Q3 + (1.5 * IQR):
Upper bound = 317.5 + (1.5 * 35) = 317.5 + 52.5 = 370
Therefore, any values below 230 and above 370 would be considered outliers in this distribution.
To find the values that would be considered outliers in this distribution, we need to use the concept of z-scores.
A z-score measures how many standard deviations an observation is from the mean. If the z-score is negative, the observation is below the mean, and if it's positive, the observation is above the mean.
In this case, we know that the z-score for the first quartile (Q1) is -0.7 and the z-score for the third quartile (Q3) is 0.7.
To calculate the actual values for Q1 and Q3, we use the formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value we are trying to find
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Rearranging the formula, we can isolate the value x:
x = μ + z * σ
Now, we can substitute the values into the formula to find the values for Q1 and Q3:
For Q1:
x1 = 300 + (-0.7) * 25 = 300 - 17.5 = 282.5
For Q3:
x3 = 300 + 0.7 * 25 = 300 + 17.5 = 317.5
So, the value for Q1 is 282.5 and the value for Q3 is 317.5.
Now, to identify outliers, we need to determine the range within which data points are typically considered non-outliers. A common method is the "1.5 * IQR rule."
The interquartile range (IQR) is the range between Q1 and Q3:
IQR = Q3 - Q1 = 317.5 - 282.5 = 35
Using the 1.5 * IQR rule, we calculate:
Lower bound outlier: Q1 - 1.5 * IQR
Upper bound outlier: Q3 + 1.5 * IQR
Lower bound outlier = 282.5 - 1.5 * 35 = 282.5 - 52.5 = 230
Upper bound outlier = 317.5 + 1.5 * 35 = 317.5 + 52.5 = 370
Therefore, any value below 230 or above 370 would be considered an outlier in this distribution.