Math Problem Statement
Consider a sample with data values of 10,20 ,12 ,17 , and 16 . Compute the z-score for each of the five observations (to 2 decimals). Enter negative values as negative numbers.
Solution
To compute the z-score for each observation, we use the following formula:
Where:
- is an individual data value.
- is the mean of the data set.
- is the standard deviation of the data set.
Step 1: Calculate the mean
Step 2: Calculate the standard deviation
First, find the squared differences from the mean for each data point:
Now, sum these squared differences:
Next, divide the sum by the number of data points minus 1 (this is the sample variance):
Finally, take the square root of the variance to get the standard deviation:
Step 3: Calculate the z-scores
Now, compute the z-score for each observation using the formula .
For :
For :
For :
For :
For :
Final z-scores (rounded to 2 decimals):
Would you like further clarification or additional details?
Here are 5 related questions:
- What does a z-score represent in statistics?
- How do z-scores relate to standard deviation?
- Why is it important to calculate z-scores in data analysis?
- Can you explain the difference between population and sample standard deviation?
- What does it mean if a z-score is negative?
Tip: Z-scores allow you to compare data points from different distributions by standardizing them.
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Math Problem Analysis
Mathematical Concepts
Statistics
Z-Score
Standard Deviation
Mean
Formulas
z = (x - μ) / σ
μ = (Σx) / n
σ = sqrt((Σ(x - μ)^2) / (n - 1))
Theorems
Z-score Theorem
Sample Standard Deviation
Suitable Grade Level
Grades 9-12
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