3. Describing Data Numerically
Standard Deviation
2:54 minutesProblem 2.Q.6a
Textbook Question
Refer to the sample statistics from Exercise 5 and determine whether any of the house prices below are unusual. Explain your reasoning.
a. $225,000
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Step 1: Recall the rule for identifying unusual values. A value is considered unusual if it lies more than 2 standard deviations away from the mean. Mathematically, this can be expressed as: \( \text{Unusual if: } x < \mu - 2\sigma \text{ or } x > \mu + 2\sigma \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.
Step 2: Obtain the mean (\( \mu \)) and standard deviation (\( \sigma \)) of the house prices from the sample statistics provided in Exercise 5. These values are necessary to calculate the range of usual values.
Step 3: Calculate the lower bound of usual values using the formula \( \mu - 2\sigma \). Substitute the mean and standard deviation into the formula to find this value.
Step 4: Calculate the upper bound of usual values using the formula \( \mu + 2\sigma \). Again, substitute the mean and standard deviation into the formula to find this value.
Step 5: Compare the given house price of $225,000 to the calculated range of usual values. If $225,000 lies outside the range \( [\mu - 2\sigma, \mu + 2\sigma] \), it is considered unusual. Otherwise, it is not unusual.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unusual Values
In statistics, an unusual value, or outlier, is a data point that significantly differs from the other observations in a dataset. Typically, values that lie beyond 1.5 times the interquartile range (IQR) from the quartiles are considered unusual. Identifying unusual values helps in understanding the distribution and variability of the data.
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Step 3: Get P-Value
Descriptive Statistics
Descriptive statistics summarize and describe the main features of a dataset. Key measures include the mean, median, mode, range, variance, and standard deviation. These statistics provide a foundation for understanding the overall trends and patterns in the data, which is essential for determining whether specific values are unusual.
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Normal Distribution
A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Understanding the properties of normal distribution, including the empirical rule (68-95-99.7), helps in assessing how likely a particular value is to be considered unusual within the context of the dataset.
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