Math Problem Statement

Below are the jersey numbers of 11 players randomly selected from a football team. Find the​ range, variance, and standard deviation for the given sample data. What do the results tell​ us? 26 98 89 57 76 24 1 83 91 45 43 Question content area bottom Part 1 Rangeequals

97 ​(Round to one decimal place as​ needed.) Part 2 Sample standard deviationequals

enter your response here ​(Round to one decimal place as​ needed.)

Solution

Let's solve this step by step.

1. Range

The range is calculated as the difference between the maximum and minimum values in the dataset.

  • Maximum value: 98
  • Minimum value: 1

Range=981=97\text{Range} = 98 - 1 = 97

This matches the range you provided.

2. Variance and Standard Deviation

The standard deviation and variance for the sample can be calculated as follows:

Step 1: Calculate the mean (average) of the data.

Mean=26+98+89+57+76+24+1+83+91+45+4311\text{Mean} = \frac{26 + 98 + 89 + 57 + 76 + 24 + 1 + 83 + 91 + 45 + 43}{11} Mean=63311=57.5455(rounded to four decimal places)\text{Mean} = \frac{633}{11} = 57.5455 \quad \text{(rounded to four decimal places)}

Step 2: Calculate each deviation from the mean and square it.

For each data point xix_i, the squared deviation is (xiMean)2(x_i - \text{Mean})^2.

Step 3: Calculate the sample variance.

Sample Variance=(xiMean)2n1\text{Sample Variance} = \frac{\sum (x_i - \text{Mean})^2}{n-1} Where nn is the number of observations.

Step 4: Calculate the sample standard deviation.

Sample Standard Deviation=Sample Variance\text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}}

Let's calculate these values.

I'll calculate these now.### Results:

  • Mean: 57.5 (rounded to one decimal place)
  • Sample Variance: 1042.1 (rounded to one decimal place)
  • Sample Standard Deviation: 32.3 (rounded to one decimal place)

Interpretation:

  • Range: The range of 97 indicates a wide spread in the jersey numbers, with values ranging from 1 to 98.
  • Variance and Standard Deviation: The high standard deviation of 32.3 shows that the jersey numbers are quite dispersed around the mean of 57.5, indicating variability in the numbers chosen by the players.

Would you like more details or have any questions?


Here are five related questions to deepen your understanding:

  1. How does the standard deviation change if the dataset is more concentrated around the mean?
  2. What is the effect of an outlier on the range and standard deviation?
  3. Can you explain why the sample variance uses n1n-1 instead of nn in the denominator?
  4. How would you interpret the standard deviation in the context of a normal distribution?
  5. How does the range compare to other measures of dispersion like interquartile range?

Tip: When calculating variance and standard deviation, always check if the data represents the entire population or just a sample. Use n1n-1 for sample data (Bessel's correction) to get an unbiased estimate.

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Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Range
Variance
Standard Deviation

Formulas

Range = Maximum value - Minimum value
Sample Variance = (sum of squared deviations) / (n-1)
Sample Standard Deviation = sqrt(Sample Variance)

Theorems

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Suitable Grade Level

Grades 9-12