Guided Reading Ch 3-1

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Jan 9, 2024

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Guided Reading Chapter 3 Fill these in to the best of your ability and submit to canvas by the deadline, using Lane https://onlinestatbook.com/2/summarizing_distributions/summarizing_distributions.html 1. What is Central Tendency? The statistical measure that identifies a single value as representative of an entire distribution. 2. What are the measures of Central Tendency? The main measures of central tendency are mean, median, mode. 3. Identify situations in which knowing the center of a distribution would be valuable It gives the value of where approximately half of your data is on one side and the rest of the data is on the other side. 4. Give three different ways the center of a distribution can be defined The three different ways the center of a distribution can be defined are mean, median, and mode. 5. Describe how the balance is different for symmetric distributions than it is for asymmetric distributions A distribution is asymmetric if it is not symmetric with zero skewness. An asymmetric distribution is either left or right skewed. 6. How do you compute a mean? You compute a mean by adding all the numbers together and divide the total sum by the number of scores used. 7. How do you compute a median? Arrange the data points from smallest to largest. If the number of data points is odd, the median is the middle point in the list. If it is even, the median is the average of the two middle data points. 8. How do you compute a mode? You place all numbers in a given set in order and count how many times each number appears in the set. 9. What does it mean to minimize the sum of absolute deviations? That means that you’re finding the median. 10. Which measure of central tendency minimizes the sum of absolute deviations? The measure of central tendency that minimizes the sum of absolute deviations is the median. 11. What does it mean to minimize the sum of squared deviations?

You take all residual, square them, and finally sum them up. 12. Which measure of central tendency minimizes the sum of square deviations? The mean is the value that minimizes all the sums of squared deviations. 13. State when the mean and median are the same. The mean and median are the same when there is a perfectly symmetrical distribution. 14. State whether it is the mean or median that minimizes the mean absolute deviation The median minimizes the mean absolute deviation. 15. State whether it is the mean or median that minimizes the mean squared deviation The mean is the value that minimizes the sum of the squared deviations. 16. State whether it is the mean or median that is the balance point on a balance scale. The total distance from the mean to the data points below the mean is equal to the entire distance from the mean to the data points which were above the mean. 17. How might the mean and median change with different distributions? If the distribution of data is skewed to the left, then that results in the mean being less than the median. 18. Define: trimean, geometric mean, and trimmed mean Trimean is the measure of central tendency that combines aspects of both the median and mean. Geometric mean is the measure of central tendency that’s calculated by multiplying all the values in a dataset and taking the nth root. The trimmed mean involves removing a certain percentage of data points from both ends of a dataset then calculating the mean. 19. How does skew affect the relationship between mean and median? When the distribution is skewed to the left the mean will be less than the median. 20. How do mean and median differ in symmetric distributions? When there is a symmetrical distribution that has two modes, these modes would be different from the mean and median. 21. State which measures of central tendency should be used to describe the center of a skewed distribution The median is preferred since it is more resistant to outliers than the mean. 22. What does variability refer to?

Variability refers to the statistical unit that creates conclusions from a data set. 23. Define the following: range, interquartile range, variance, spread, dispersion. The range represents the difference between the maximum and minimum values in a dataset. The interquartile range focuses on the center of the dataset. The variance quantifies the average squared difference between each data point. The spread refers to how data points are distributed across the range of values. 24. How are the variance of a sample and variance of a population related? If all kinds of observations of the system are present then the calculated variance is the population variance. 25. How do you calculate standard deviation from variance? You have to calculate the square root of the variance. 26. How do changes in the mean and standard deviation affect the position and shape of the distribution? The changing of the mean shifts the curve along the horizontal axis without changing its shape. 27. How do you calculate skew? You calculate the skew by multiplying 3 by the mean subtracted from the median and divided that by the standard deviation. 28. How do you calculate kurtosis? Kurtosis is calculated by using the formula β 2 = (E(x) 4 / (E(x) 2 ) 2 ) − 3, 29. Define a linear transformation A linear transformation is a function from one vector space to another that respects the linear structure of each vector. 30. How do you compute the variance of the sum of two uncorrelated (independent) variables? You would use the formula Var[X+Y]=Var[X] + Var[Y] +2(E[XY]-E[X]*E[Y]). 31. How do you compute the variance of the difference between two uncorrelated (independent) variables? To compute the variance of the difference between two independent variables, you would add the individual variances of the two.

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Guided Reading Ch 3-1

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