Math Problem Statement

4) Consider the dataset: 3, 4, 1, 2, 5. The Z-score of the number 5 is given by: 5) Consider the following dataset: 17, 19, 36, 18, 9, 11, 21, 14, 16, 14. The number of outliers for this dataset is equal to:

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: Z=xμσZ = \frac{x - \mu}{\sigma} where:

  • xx is the data point (in this case, 5),
  • μ\mu is the mean of the dataset,
  • σ\sigma is the standard deviation of the dataset.

Step 1: Calculate the mean (μ\mu)

μ=3+4+1+2+55=155=3\mu = \frac{3 + 4 + 1 + 2 + 5}{5} = \frac{15}{5} = 3

Step 2: Calculate the standard deviation (σ\sigma)

First, find the squared differences from the mean:

  • (33)2=0(3 - 3)^2 = 0
  • (43)2=1(4 - 3)^2 = 1
  • (13)2=4(1 - 3)^2 = 4
  • (23)2=1(2 - 3)^2 = 1
  • (53)2=4(5 - 3)^2 = 4

Now, find the variance (average of squared differences): Variance=0+1+4+1+45=105=2\text{Variance} = \frac{0 + 1 + 4 + 1 + 4}{5} = \frac{10}{5} = 2 Then, the standard deviation is the square root of the variance: σ=21.41\sigma = \sqrt{2} \approx 1.41

Step 3: Calculate the Z-score

Now, apply the Z-score formula for x=5x = 5: Z=531.41=21.411.42Z = \frac{5 - 3}{1.41} = \frac{2}{1.41} \approx 1.42

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 Q11.5×IQRQ1 - 1.5 \times \text{IQR} or above Q3+1.5×IQRQ3 + 1.5 \times \text{IQR}.

Step 1: Arrange the data in ascending order

9,11,14,14,16,17,18,19,21,369, 11, 14, 14, 16, 17, 18, 19, 21, 36

Step 2: Find the quartiles

  • Q1Q1 (25th percentile) is the median of the lower half: Q1=14Q1 = 14,
  • Q3Q3 (75th percentile) is the median of the upper half: Q3=19Q3 = 19.

Step 3: Calculate the IQR

IQR=Q3Q1=1914=5\text{IQR} = Q3 - Q1 = 19 - 14 = 5

Step 4: Find the outlier bounds

Lower bound=Q11.5×IQR=141.5×5=147.5=6.5\text{Lower bound} = Q1 - 1.5 \times \text{IQR} = 14 - 1.5 \times 5 = 14 - 7.5 = 6.5 Upper bound=Q3+1.5×IQR=19+1.5×5=19+7.5=26.5\text{Upper bound} = Q3 + 1.5 \times \text{IQR} = 19 + 1.5 \times 5 = 19 + 7.5 = 26.5

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:

  1. How do you calculate the Z-score for a negative data point?
  2. What happens to the standard deviation if all data points in a dataset increase by the same constant?
  3. How is variance related to standard deviation?
  4. Can a dataset have multiple Z-scores for the same value?
  5. 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)