Question

Find the mean the standard deviation, and the z-scores corresponding to the minimum and maximum in the data set. Do the z-scores indicate that there are possible outliers in these data sets? \(n=13\) measurements: 3,9,10,2,6,7,5,8,6,6,4,9,25

Short answer

Answer: Yes, there is one possible outlier in the data set. The maximum value of 25 is considered an outlier since its z-score (4.34) is farther than 2 standard deviations from the mean.

Step-by-step solution

Calculate the mean of the data set

To calculate the mean, we need to sum all the data points, and divided the result by the number out data points (n=13). Mean = \(\frac{3+9+10+2+6+7+5+8+6+6+4+9+25}{13} = \frac{100}{13} = 7.69\)

Calculate the standard deviation of the data set

To calculate the standard deviation, we will use the following formula: Standard deviation = \(\sqrt{\frac{\Sigma(x_i - \bar{x})^2}{n}}\) where \(x_i\) are the data points, \(\bar{x}\) is the mean, and n is the number of data points. Subtract the mean from each data point, square the results, and sum them: \((3-7.69)^2+(9-7.69)^2+(10-7.69)^2+(2-7.69)^2+(6-7.69)^2+(7-7.69)^2+(5-7.69)^2+(8-7.69)^2+(6-7.69)^2+(6-7.69)^2+(4-7.69)^2+(9-7.69)^2+(25-7.69)^2=165.38\) Divide the sum by 13 and take the square root: Standard deviation = \(\sqrt{\frac{165.38}{13}} = 3.98\)

Find the minimum and maximum values in the data set

The minimum value in the data set is 2, and the maximum value in the data set is 25.

Calculate the z-scores for the minimum and maximum values

The formula for calculating the z-score is: Z-score = \(\frac{x - \bar{x}}{\sigma}\) where x is the data point, \(\bar{x}\) is the mean, and \(\sigma\) is the standard deviation. Z-score (minimum) = \(\frac{2 - 7.69}{3.98} = -1.43\) Z-score (maximum) = \(\frac{25 - 7.69}{3.98} = 4.34\)

Analyze the z-scores for possible outliers

A z-score close to 0 indicates that the data point is not far from the mean, while a z-score that is larger in magnitude (either positive or negative) indicates that the data point is farther from the mean. Outliers are typically considered data points with z-scores farther than 2 or -2 standard deviations from the mean. In this case, our minimum z-score of -1.43 is within 2 standard deviations of the mean, so it is not considered an outlier. However, our maximum z-score of 4.34 is farther than 2 standard deviations from the mean, so it is considered a possible outlier.

Key concepts

Standard Deviation

Standard deviation is a crucial statistic that measures the amount of variation or dispersion in a set of values. To put it simply, it tells us how spread out the numbers in a data set are. In the example provided, we see how standard deviation is calculated step-by-step. First, the mean, or average, of the data set is determined. Each data point is then subtracted from this mean to find the difference. These differences are squared to eliminate negative values and then summed up. The result divided by the number of data points gives the variance, and the square root of the variance is the standard deviation.

Understanding standard deviation is vital as it helps identify how consistent the data points are. A smaller standard deviation indicates that the values are closer to the mean, while a larger one suggests greater variation. This information is incredibly beneficial when comparing different data sets or assessing the reliability of the data.

Outliers in Data Sets

Outliers are data points that differ significantly from other observations in a data set. They can arise due to measurement variability, experimental errors, or possibly indicate a novel finding. In the exercise, the concept of outliers is explored by calculating z-scores for the maximum and minimum values in the data set.

A z-score helps determine the position of a data point relative to the mean, and it is measured in terms of standard deviations. If a data point's z-score is beyond a certain threshold (commonly set at 2 or -2), it may be labeled as an outlier. As illustrated, the maximum value in our data set has a z-score of 4.34, which is well above the threshold, suggesting it's an outlier. Being aware of outliers is important because they can have a substantial impact on statistical analyses, such as skewing the mean or affecting the results of predictive modeling.

Probability and Statistics

Probability and statistics are branches of mathematics that deal with data analysis and the prediction of events. While probability addresses the likelihood of an event occurring, statistics involves collecting, analyzing, interpreting, presenting, and organizing data. Z-scores are a link between the two, allowing us to interpret individual data points in the context of the entire data set.

In our textbook exercise, after calculating the z-scores, we can infer the probability of finding similar or more extreme values in the same data set. This approach is fundamental in statistical hypothesis testing, where we use probability to make inferences about a population, based on sample data. Understanding these concepts will not only help in solving homework problems but also in making informed decisions based on data in real-world circumstances.