Question

measures of dispersion
ten friends kept track of how many times they sent a text over a one hour period. the results are in the table below.
number of texts | 3 | 4 | 5 | 5 | 6 | 6 | 12 | 13 | 18 | 20
they determine that the mean of the data set is 9.2, but they also want to know the range and standard deviation.
calculate the range of the data set.
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calculate the sample standard deviation of the data set.
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round your result to the two decimal places as needed.
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Explanation:

Part 1: Calculate the range of the data set

Step1: Identify max and min values

The data set is \( 3, 4, 5, 5, 6, 6, 12, 13, 18, 20 \). The maximum value (\( \text{max} \)) is \( 20 \), and the minimum value (\( \text{min} \)) is \( 3 \).

Step2: Calculate the range

The formula for the range is \( \text{Range} = \text{max} - \text{min} \). Substituting the values, we get \( \text{Range} = 20 - 3 = 17 \).

Step1: Recall the formula for sample standard deviation

The formula for the sample standard deviation (\( s \)) is \( s = \sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}} \), where \( x_{i} \) are the data points, \( \bar{x} \) is the mean, and \( n \) is the number of data points. Here, \( \bar{x}=9.2 \) and \( n = 10 \).

Step2: Calculate \( (x_{i}-\bar{x})^{2} \) for each data point

• For \( x_{1}=3 \): \( (3 - 9.2)^{2}=(-6.2)^{2}=38.44 \)
• For \( x_{2}=4 \): \( (4 - 9.2)^{2}=(-5.2)^{2}=27.04 \)
• For \( x_{3}=5 \): \( (5 - 9.2)^{2}=(-4.2)^{2}=17.64 \)
• For \( x_{4}=5 \): \( (5 - 9.2)^{2}=(-4.2)^{2}=17.64 \)
• For \( x_{5}=6 \): \( (6 - 9.2)^{2}=(-3.2)^{2}=10.24 \)
• For \( x_{6}=6 \): \( (6 - 9.2)^{2}=(-3.2)^{2}=10.24 \)
• For \( x_{7}=12 \): \( (12 - 9.2)^{2}=(2.8)^{2}=7.84 \)
• For \( x_{8}=13 \): \( (13 - 9.2)^{2}=(3.8)^{2}=14.44 \)
• For \( x_{9}=18 \): \( (18 - 9.2)^{2}=(8.8)^{2}=77.44 \)
• For \( x_{10}=20 \): \( (20 - 9.2)^{2}=(10.8)^{2}=116.64 \)

Step3: Sum up the squared deviations

\( \sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=38.44 + 27.04+17.64 + 17.64+10.24 + 10.24+7.84 + 14.44+77.44 + 116.64 \)
\( = 38.44+27.04 = 65.48 \); \( 65.48+17.64 = 83.12 \); \( 83.12+17.64 = 100.76 \); \( 100.76+10.24 = 111 \); \( 111+10.24 = 121.24 \); \( 121.24+7.84 = 129.08 \); \( 129.08+14.44 = 143.52 \); \( 143.52+77.44 = 220.96 \); \( 220.96+116.64 = 337.6 \)

Step4: Calculate the variance (sample)

The sample variance (\( s^{2} \)) is \( \frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}=\frac{337.6}{10 - 1}=\frac{337.6}{9}\approx37.5111 \)

Step5: Calculate the sample standard deviation

Take the square root of the sample variance: \( s=\sqrt{37.5111}\approx6.12 \) (rounded to two decimal places)

Answer:

\( 17 \)

Part 2: Calculate the sample standard deviation of the data set