Math Problem Statement
Solution
Let's address the two questions in the image:
4. Consider the dataset: 3, 4, 1, 2, 5. The Z-score of the number 5 is given by:
To calculate the Z-score, we use the formula: where:
- is the data point (in this case, 5),
- is the mean of the dataset,
- is the standard deviation of the dataset.
Step 1: Calculate the mean ()
Step 2: Calculate the standard deviation ()
First, find the squared differences from the mean:
Now, find the variance (average of squared differences): Then, the standard deviation is the square root of the variance:
Step 3: Calculate the Z-score
Now, apply the Z-score formula for :
This is closest to Option d) 1.58.
5. Consider the following dataset: 17, 19, 36, 18, 9, 11, 21, 14, 16, 14. The number of outliers for this dataset is:
To identify the outliers, we use the interquartile range (IQR) method. Outliers are data points that fall below or above .
Step 1: Arrange the data in ascending order
Step 2: Find the quartiles
- (25th percentile) is the median of the lower half: ,
- (75th percentile) is the median of the upper half: .
Step 3: Calculate the IQR
Step 4: Find the outlier bounds
Step 5: Identify outliers
Any value below 6.5 or above 26.5 is considered an outlier. In the dataset, 36 is the only outlier.
Therefore, the number of outliers is 1 (Option b).
Would you like more details on these calculations or any clarification?
Related Questions:
- How do you calculate the Z-score for a negative data point?
- What happens to the standard deviation if all data points in a dataset increase by the same constant?
- How is variance related to standard deviation?
- Can a dataset have multiple Z-scores for the same value?
- How do different methods (e.g., IQR, Z-score) compare when identifying outliers?
Tip:
When working with datasets, always organize the data in ascending order before calculating quartiles and IQR for outlier detection.
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Math Problem Analysis
Mathematical Concepts
Statistics
Z-Score
Interquartile Range (IQR)
Outliers
Formulas
Z-score formula: Z = (x - μ) / σ
Standard deviation (σ)
Interquartile range (IQR): IQR = Q3 - Q1
Outlier bounds: Q1 - 1.5 * IQR and Q3 + 1.5 * IQR
Theorems
The Empirical Rule for Z-scores
IQR Method for Detecting Outliers
Suitable Grade Level
Grades 10-12 (High School)