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? \(\begin{aligned} n=11 \text { measurements: } & 2.3,1.0,2.1,6.5,2.8,7.8, \\ & 1.7,2.9,4.4,5.1,2.0 \end{aligned}\)

Short answer

Answer: No, there are no outliers in the data set, as the z-scores for both the minimum and maximum measurements lie between -2 and 2.

Step-by-step solution

Calculate the mean

Add all the measurements and divide the sum by the total number of measurements (n=11). Mean = (2.3 + 1.0 + 2.1 + 6.5 + 2.8 + 7.8 + 1.7 + 2.9 + 4.4 + 5.1 + 2.0)/11 = 38.6/11 = 3.51

Find the differences from the mean

Subtract the mean from each measurement: 2.3 - 3.51 = -1.21 1.0 - 3.51 = -2.51 2.1 - 3.51 = -1.41 6.5 - 3.51 = 2.99 2.8 - 3.51 = -0.71 7.8 - 3.51 = 4.29 1.7 - 3.51 = -1.81 2.9 - 3.51 = -0.61 4.4 - 3.51 = 0.89 5.1 - 3.51 = 1.59 2.0 - 3.51 = -1.51

Calculate the mean of squared differences

Square each difference, then find the average of those squared differences: (-1.21)^2 + (-2.51)^2 + (-1.41)^2 + 2.99^2 + (-0.71)^2 + 4.29^2 + (-1.81)^2 + (-0.61)^2 + 0.89^2 + 1.59^2 + (-1.51)^2 = 54.3451 Mean of squared differences = 54.3451/11 = 4.94046

Find the standard deviation

Take the square root of the mean of squared differences: Standard deviation = \(\sqrt{4.94046} \approx 2.22\)

Find z-scores for minimum and maximum measurements

Minimum measurement is 1.0, and maximum measurement is 7.8. Subtract the mean from each of these and divide by the standard deviation to find their z-scores: Z-score minimum = (1.0 - 3.51) / 2.22 = -1.13 Z-score maximum = (7.8 - 3.51) / 2.22 = 1.93

Determine if z-scores indicate outliers

Since both z-scores lie between -2 and 2, they do not indicate any outliers in these data sets.

Key concepts

Mean Calculation

The mean, also known as the average, is a basic concept in descriptive statistics. It's used to find the central tendency of a dataset. To calculate the mean, add up all the numbers in your dataset. In our example, the numbers are: 2.3, 1.0, 2.1, 6.5, 2.8, 7.8, 1.7, 2.9, 4.4, 5.1, and 2.0.
Add all these values to get 38.6. Then, divide this sum by the number of values, which is 11. This calculation yields a mean of approximately 3.51.
The mean provides a simpler way to understand the overall 'typical' value of the dataset, meaning each data point would hypothetically equal this value if they were evenly distributed.

Standard Deviation

The standard deviation is a critical measure in statistics that describes the amount of variation or dispersion in a dataset.
It tells us how spread out the numbers are around the mean.
To find the standard deviation:

• First, calculate the deviation of each number from the mean (subtract the mean from each value).
• Next, square each deviation to make them positive and emphasize larger deviations.
• Then, find the average of these squared values.
• Finally, take the square root of this average to get the standard deviation.

In the given dataset, the standard deviation comes out to be about 2.22.
This value shows the average spread of data points from the mean, helping us understand how closely the data are clustered around their mean value.

Z-scores

Z-scores are a way of standardizing data points within a dataset. They tell us how many standard deviations a data point is from the mean.
To calculate a z-score for a value, subtract the mean from the value and divide by the standard deviation:

• For the minimum value of 1.0: \[ Z = \frac{1.0 - 3.51}{2.22} \approx -1.13 \]
• For the maximum value of 7.8: \[ Z = \frac{7.8 - 3.51}{2.22} \approx 1.93 \]

Z-scores allow us to compare data points from different datasets effectively.
They also aid in determining how typical or atypical certain values are relative to the dataset. In this case, these z-scores do not suggest extreme values.

Outlier Detection

Detecting outliers is essential in data analysis because they can skew results and affect statistical inferences.
Outliers are typically identified using z-scores, with a common rule of thumb being that any z-score over 2 or under -2 might indicate a potential outlier.
In this dataset:

• The minimum value's z-score is -1.13.
• The maximum value's z-score is 1.93.

Both of these values fall between -2 and 2, indicating that there are no significant outliers.
Understanding and identifying outliers is important for accurate data interpretation, as they can be anomalies that either provide valuable insights or represent data errors.