3 Goalscalculate The Arithmetic Mean Weighted Mean Median Mode An ✓ Solved

3-* GOALS Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean. Explain the characteristics, uses, advantages, and disadvantages of each measure of location. Identify the position of the mean, median, and mode for both symmetric and skewed distributions. Compute and interpret the range, mean deviation, variance, and standard deviation. Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion.

Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations. * 3-* Parameter Versus Statistics PARAMETER A measurable characteristic of a population. STATISTIC A measurable characteristic of a sample. * 3-* Population Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values. The sample mean is the sum of all the sample values divided by the total number of sample values. EXAMPLE: * 3-* The Median PROPERTIES OF THE MEDIAN There is a unique median for each data set. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.

It can be computed for ratio-level, interval-level, and ordinal-level data. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. EXAMPLES: The ages for a sample of five college students are: 21, 25, 19, 20, 22 Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. The heights of four basketball players, in inches, are: 76, 73, 80, 75 Arranging the data in ascending order gives: 73, 75, 76, 80.

Thus the median is 75.5 MEDIAN The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest. * 3-* The Mode MODE The value of the observation that appears most frequently. * 3-* The Relative Positions of the Mean, Median and the Mode * 3-* The Geometric Mean Useful in finding the average change of percentages, ratios, indexes, or growth rates over time. It has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the GDP, which compound or build on each other. The geometric mean will always be less than or equal to the arithmetic mean.

The formula for the geometric mean is written: EXAMPLE: Suppose you receive a 5 percent increase in salary this year and a 15 percent increase next year. The average annual percent increase is 9.886, not 10.0. Why is this so? We begin by calculating the geometric mean. * 3-* Measures of Dispersion A measure of location, such as the mean or the median, only describes the center of the data. It is valuable from that standpoint, but it does not tell us anything about the spread of the data.

For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to wade across on foot without additional information? Probably not. You would want to know something about the variation in the depth. A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions. RANGE MEAN DEVIATION VARIANCE AND STANDARD DEVIATION * 3-* EXAMPLE – Mean Deviation EXAMPLE: The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80.

Determine the mean deviation for the number of cappuccinos sold. Step 1: Compute the mean Step 2: Subtract the mean (50) from each of the observations, convert to positive if difference is negative Step 3: Sum the absolute differences found in step 2 then divide by the number of observations * 3-* Variance and Standard Deviation The variance and standard deviations are nonnegative and are zero only if all observations are the same. For populations whose values are near the mean, the variance and standard deviation will be small. For populations whose values are dispersed from the mean, the population variance and standard deviation will be large. The variance overcomes the weakness of the range by using all the values in the population VARIANCE The arithmetic mean of the squared deviations from the mean.

STANDARD DEVIATION The square root of the variance. * 3-* EXAMPLE – Population Variance and Population Standard Deviation The number of traffic citations issued during the last five months in Beaufort County, South Carolina, is reported below: What is the population variance? Step 1: Find the mean. Step 2: Find the difference between each observation and the mean, and square that difference. Step 3: Sum all the squared differences found in step 3 Step 4: Divide the sum of the squared differences by the number of items in the population. * 3-* Sample Variance and Standard Deviation EXAMPLE The hourly wages for a sample of part-time employees at Home Depot are: , , , , and . What is the sample variance? * 3-* Chebyshev’s Theorem and Empirical Rule The arithmetic mean biweekly amount contributed by the Dupree Paint employees to the company’s profit-sharing plan is .54, and the standard deviation is .51.

At least what percent of the contributions lie within plus 3.5 standard deviations and minus 3.5 standard deviations of the mean? * 3-* EXAMPLE: Determine the arithmetic mean vehicle selling price given in the frequency table below. The Arithmetic Mean and Standard Deviation of Grouped Data EXAMPLE Compute the standard deviation of the vehicle selling prices in the frequency table below. * . ) . )( . ( GM = = = + + + + = = ॠn x x , = = - = ॠN X m s ... = = + + + + = = ॠN x m sample the in ns observatio of number the is sample the of mean the is sample the in n observatio each of value the is variance sample the is : Where 2 n X X s Course Scenario On March 12, 2014, at approximately 2200 hours, the Sunnyville, Utah Police Department received a 911 call of an armed robbery at 201 SE 2nd Ave.

Upon the police arriving on scene, Victim 1, Luke Roberts, had been shot in the head and deceased. Victim 2, Liam O'Neil, was pistol-whipped. Both victims were robbed of their cell phones and wallets at gunpoint. Liam O'Neil was able to identify both suspects and the getaway vehicle. A short distance away, police stopped the suspect vehicle with two Caucasian males that matched the description provided by Mr.

O'Neil. After the two suspects were positively identified, they were arrested and brought to the police station for interviews. Before the police interviewed the suspects, they read them their Miranda rights. Suspect 1 refused to speak to the police, invoked his Miranda rights, and stated that he wanted a lawyer. The police began asking Suspect 2 specific questions about the crime.

Suspect 2 stated “I’m not sure I should talk to you,†but then hesitantly proceeded to answers questions, making several incriminating statements. The police conducted a second interview. As Suspect 2, Keith Hopkins, waited for the detective to speak to him again, he appeared to be deleting messages from his phone. As the second interview began, Hopkins admitted to hiding the stolen property from both victims inside his residence located at 1106 SE 9th Ave. He stated that the victim's property was hidden in a laundry basket in his bedroom.

He also stated that Suspect 1, Steve Chapman, hid the gun used in the robbery in the attic of his house under the insulation. You suspected that there may also be additional evidence in the house, but Keith Hopkins will no longer provide you with any information. The police asked Mr. Hopkins for consent to search his residence, but he immediately remembered an episode of his favorite police show and refused to give you permission to search his house for the evidence. After completing a thorough investigation into the robbery and serving your search warrant, both of your suspects were found guilty at trial.

The next step of the Criminal Justice System (the sentencing phase) will begin. Because victim Roberts was shot in the head and killed, the State is seeking the death penalty for Steve Chapman. He has an extensive violent criminal history (convicted felon), and shows no remorse for victim Roberts or his family. As the lead detective, you have completed your investigation into the robbery and homicide. You have served the search warrant and found all of the evidence that you were looking for.

You and your team have collected all of the evidence and interviewed witnesses, the victim, and the suspects. Both suspects had a lengthy history with drugs and alcohol. Both suspects have been in drug treatment several times. They have had no success in recovering from their addictions. It was also determined that both suspects were under the influence of drugs and alcohol at the time of the robbery.

Suspect Hopkins is now being sentenced to life in prison for his involvement in the robbery. The State’s Attorney has taken into consideration Keith Hopkins’ cooperation in the investigation, as well as his remorse for the victims.

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Statistical Measures of Central Tendency and Dispersion

In statistics, measures of central tendency such as arithmetic mean, weighted mean, median, mode, and geometric mean, along with measures of dispersion like range, variance, mean deviation, and standard deviation, play a pivotal role in data analysis. This discussion will cover how to calculate these statistics and explain their characteristics, uses, advantages, and disadvantages. Furthermore, we will delve into the relationships of these measures with various distributions, Chebyshev’s theorem, and the Empirical Rule.

Measures of Central Tendency

1. Arithmetic Mean
- Calculation: The arithmetic mean is calculated as the sum of all values divided by the total number of observations. For instance, the mean of values \(x_1, x_2, \ldots, x_n\) is given by:
\[
\text{Mean} = \frac{x_1 + x_2 + \ldots + x_n}{n}
\]
- Characteristics: Sensitive to extreme values (outliers). It is the most common measure of central tendency that incorporates all data points.
- Advantages: Simple to calculate and understand.
- Disadvantages: Misleading when the data set contains outliers, as it can be skewed by very high or very low values.
2. Weighted Mean
- Calculation: Used when different data points contribute unequally to the average. The weighted mean can be computed as:
\[
\text{Weighted Mean} = \frac{(w_1 \times x_1) + (w_2 \times x_2) + \ldots + (w_n \times x_n)}{w_1 + w_2 + \ldots + w_n}
\]
where \(w_i\) are the weights assigned to each data point.
- Characteristics: Takes into account the importance of each data point.
- Advantages: More accurate representation where certain data points are more significant.
- Disadvantages: Requires the selection of appropriate weights, which can introduce bias.
3. Median
- Calculation: The median is the middle value when data is ordered from smallest to largest. For a dataset with \(n\) observations:
- If \(n\) is odd, the median is \(x_{(\frac{n+1}{2})}\).
- If \(n\) is even, it is calculated as \(\frac{x_{(\frac{n}{2})} + x_{(\frac{n}{2}+1)}}{2}\).
- Characteristics: Not impacted by outliers.
- Advantages: Ideal for skewed distributions or when outliers exist.
- Disadvantages: Does not utilize all data points, potentially leading to loss of information.
4. Mode
- Calculation: The mode is the value that appears most frequently in the dataset.
- Characteristics: There can be more than one mode (bimodal, multimodal) or no mode at all if all values are unique.
- Advantages: Simple to identify and useful for categorical data.
- Disadvantages: May not exist or may not be unique.
5. Geometric Mean
- Calculation: Useful for datasets with multiplicative relationships, given by:
\[
GM = (x_1 x_2 ... * x_n)^{\frac{1}{n}}
\]
- Characteristics: Always less than or equal to the arithmetic mean.
- Advantages: Suitable for growth rates and percentages.
- Disadvantages: Cannot be computed if any value is non-positive.

Measures of Dispersion

1. Range
- Calculation: It is the difference between the maximum and minimum values in a dataset.
\[
\text{Range} = x_{\text{max}} - x_{\text{min}}
\]
- Advantages: Easy to compute.
- Disadvantages: Sensitive to outliers.
2. Mean Deviation
- Calculation: The average of absolute differences from the mean.
\[
\text{Mean Deviation} = \frac{\sum |x_i - \text{Mean}|}{n}
\]
- Advantages: Offers insight into average distance from the mean without regard to direction.
- Disadvantages: Like the mean, it may be influenced by extreme values.
3. Variance and Standard Deviation
- Variance: The average of the squared differences from the mean.
\[
\sigma^2 = \frac{\sum (x_i - \text{Mean})^2}{n} \quad \text{(for population)}
\]
- Standard Deviation: The square root of variance, indicating the average distance of data points from the mean.
\[
\sigma = \sqrt{\sigma^2}
\]
- Advantages: Utilizes all data points and provides information about the overall spread.
- Disadvantages: Sensitive to outliers and can be complex to interpret.

Relationships in Distributions

- Symmetric Distribution: In a perfectly symmetric distribution, the mean, median, and mode coincide at the center of the distribution (e.g., normal distribution).
- Skewed Distribution: In a right-skewed distribution, the mean is greater than the median, and both are greater than the mode. Conversely, in a left-skewed distribution, the mean is less than the median.

Chebyshev’s Theorem and Empirical Rule

- Chebyshev’s Theorem applies to all distributions, asserting that for \(k\) standard deviations from the mean, at least \((1 - \frac{1}{k^2})\) of observations lie within that range.
- Empirical Rule is applicable to normal distributions, suggesting that approximately:
- 68% of observations lie within 1 standard deviation of the mean.
- 95% within 2 standard deviations.
- 99.7% within 3 standard deviations.

Conclusion

In summary, the measures of central tendency and dispersion are vital for summarizing datasets effectively. Each measure has its own strengths and weaknesses, and their suitability depends on the underlying data characteristics, such as the presence of outliers or the nature of the data distribution. Understanding how to apply these statistical tools is essential for meaningful data analysis and interpretation (Freedman et al., 2007; Field, 2013; Moore et al., 2013).

References

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2. Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage Publications.
3. Moore, D. S., McCabe, G. P., & Craig, B. A. (2013). Introduction to the Practice of Statistics (7th ed.). W.H. Freeman.
4. Triola, M. F. (2018). Elementary Statistics (13th ed.). Pearson.
5. Bluman, A. G. (2017). Elementary Statistics: A Step by Step Approach (10th ed.). McGraw-Hill Education.
6. Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2013). Introduction to Probability and Statistics (14th ed.). Cengage Learning.
7. Sullivan, M. (2019). Statistics (6th ed.). Pearson.
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