To calculate the percentage of data points that fall within one standard deviation of the mean for the given data set {50, 46, 54, 51, 29, 52, 48, 54, 47, 48}, follow these steps:
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Calculate the mean: \[ \text{Mean} = \frac{50 + 46 + 54 + 51 + 29 + 52 + 48 + 54 + 47 + 48}{10} = \frac{481}{10} = 48.1 \]
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Calculate the standard deviation:
- First, find the deviations from the mean for each data point, square them, and calculate the variance: \[ \text{Deviations} = (50 - 48.1)^2, (46 - 48.1)^2, (54 - 48.1)^2, (51 - 48.1)^2, (29 - 48.1)^2, (52 - 48.1)^2, (48 - 48.1)^2, (54 - 48.1)^2, (47 - 48.1)^2, (48 - 48.1)^2 \] \[ = (1.9)^2, (-2.1)^2, (5.9)^2, (2.9)^2, (-19.1)^2, (3.9)^2, (-0.1)^2, (5.9)^2, (-1.1)^2, (-0.1)^2 \] \[ = 3.61, 4.41, 34.81, 8.41, 364.81, 15.21, 0.01, 34.81, 1.21, 0.01 \]
\[ \text{Variance} = \frac{3.61 + 4.41 + 34.81 + 8.41 + 364.81 + 15.21 + 0.01 + 34.81 + 1.21 + 0.01}{10} = \frac{473.3}{10} = 47.33 \] \[ \text{Standard Deviation} = \sqrt{47.33} \approx 6.88 \]
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Find the range within one standard deviation of the mean:
- Mean ± Standard Deviation: \[ \text{Lower Bound} = 48.1 - 6.88 \approx 41.22 \] \[ \text{Upper Bound} = 48.1 + 6.88 \approx 54.98 \]
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Count the data points within that range:
- Data points: 50, 46, 54, 51, 29, 52, 48, 54, 47, 48
- Points within [41.22, 54.98]: {50, 46, 54, 51, 52, 48, 54, 47, 48} (9 data points)
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Calculate the percentage: \[ \text{Percentage} = \left(\frac{9}{10}\right) \times 100 = 90% \]
The expected percentage of data points falling within one standard deviation of the mean in a normal distribution is approximately 68%.
Result: 90%; This percentage is higher than expected in a normal distribution.