PSY Ch. 3 Problem Set
#1 Introduction to measures of central tendency 1. Which of the following are measures of central tendency? 2. The median is ... 3. The equation for a sample mean is:
1. A population parameter , the median & the highest value. 2. The midpoint of the ordered scores. 3. M = Sigma X / n
The mean has a balance point
1. Complete the following table for the five values that are displayed on the graph. x = 0 : 4 points below the mean x = 0 : 4 points below the mean x = 6 : 2 points above the mean x = 7 : 3 points above the mean x = 7 : 3 points above the mean 2. Complete the following statements for thhe 5 values that are displayed on the graph. The total distance below the mean is 8 points. The total distance above the mean is 8 points. 3. Drag the orange box over the green "M" triangle is.
#10 Calculating the mean, median, and mode when scores are changed or removed. 1. Consider the following sample set of scores. Assume these scores are from a discrete distribution. 1 22 40 45 45 52 60 69 81 94 2. Suppose the score 1 in the data is mistakenly recorded as 21 instead of 1. 3. Suppose the score 1 in the original sample is inadvertently removed from the sample.
1. For the given data, the mean is 50.9 , the median is 48.5 , and the mode is 45 2. For the sample with this error, the mean is 52.9 , the median is 48.5 , and the mode is 45 . The mean increases , the median remains the same , and the mode remains the same 3. For the sample with this score removed, the mean is 50.8 , the median is 48.5 , and the mode is 45 . The mean decreases , the median remains the same , and the mode remains the same .
#11 Understanding the different measures of central tendency (2nd attempt) The mean, median, and mode are all measures of central tendency. 1. mean 2. median 3. mode
1. It always corresponds to an actual score in the data ; It is easily influenced by extreme scores ; it is algebraically defined. 2. It can be found for an open-ended distribution 3. It is the most frequently occuring score ; it is also referred to as the arithmetic average ; it can be used for data that are measured on a nominal scale.
#11 Understanding the different measures of central tendency The mean, median, and mode are all measures of central tendency. The mean is usually the preferred measure of central tendency, but there are specific situations in which it is impossible to compute a mean or in which the mean is not particularly representative of the distribution. 1. Which of the following statements about the mean are true? Check all that apply. 2. Which of the following statements about the median are true? Check all that apply. 3. Which of the following statements about the mode are true? Check all that apply.
1. It is algebraically defined; there can be more than one ; it can be used for data that are measured on a nominal scale. 2. It can be found for an open-ended distribution ; it is algebraically defined 3. It is algebraically defined ; there can be more than one ; it is the score at the 50th percentile.
#1 Introduction to measures of central tendency (2nd attempt) 1. Which of the following are measures of central tendency? 2. The mean is ... 3. The equation for a population mean is:
1. The median 2. the arithmetic average of all the scores 3. u = Sigma X / N
#1 Introduction to measures of central tendency (3rd attempt) 1. Which of the following are measures of central tendency? 2. The mode is ... 3. The equation for a population mean is:
1. The median , a population parameter, and the highest value 2. the score with the greatest frequency 3. u = Sigma X / N
#8 Computing the median for discrete and continuous variables 1. Consider the sample of n = 5 scores whose frequency distribution histogram is shown here. 2. Assume that the scores are measurements of a discrete variable. Find the median. 3. Now assume that the scores are measurements of a continuous variable. Find the precise median.
1. The total area in the histogram =5 boxes. Half the area in the histogram = 2.5 boxes Put the vertical line at 2.5 which is a little bit before the number 3 on the horizontal line. 2. The median is 3. 3. The median is 2.75
#8 Computing the median for discrete and continuous variables (2nd attempt) 1. Consider the sample of n = 6 scores whose frequency distribution histogram is shown here. 2. Assume that the scores are measurements of a discrete variable. Find the median. 3. Now assume that the scores are measurements of a continuous variable. Find the precise median.
1. The total area in the histogram =6 boxes. Half the area in the histogram = 3 boxes Put the vertical line at 2.5 which is a little bit before the number 3 on the horizontal line. 2. The median is 3. 3. The median is 2.75
The weighted mean u = (N x u) + (N x u) / N1 + N2
1. Without calculating the weighted mean for the combined group, you know that the weighted mean is: Closer to 10 than to 16 2. Compute the weighted mean. Enter your answer rounded to one decimal place: u = 12.5 3. Compute the weighted mean. Enter your answer rounded to one decimal place : u = 15.5
The weighted mean (2nd attempt) u = (N x u) + (N x u) / N1 + N2
1. Without calculating the weighted mean for the combined group, you know that the weighted mean is: Closer to 12 than to 14 2. Compute the weighted mean. Enter your answer rounded to one decimal place: u = 12.8 3. Compute the weighted mean. Enter your answer rounded to one decimal place : u = 13.6
#9 Finding the mode (2nd attempt) Consider the sample of n = 7 scores whose frequency distribution histogram is shown here. 1. The histogram is: 2. Place a purple diamond above the mode, or above each mode, if there is more than one. (Note: To receive credit, you must drag the diamond(s) into the center of the box directly above the mode(s) you want to mark.)
1. bimodal 2. The diamonds should go in the coordinate (1,3) and (4,3)
#7 Calculating the mean when multiplying or dividing by a constant 1. A school district is deciding whether or not to adopt a new math curriculum. Three classrooms are randomly selected as test sites for the curriculum. To obtain a measure of student improvement, students in the three classrooms are tested on their math skills before and after the curriculum is implemented. The math skills test has 25 possible points. The post-implementation scores for the students in the third classroom are shown in the following table: 19 13 16 18 15 14 17 8 20 19 22 11. Calculate the sample size, n; the total sum, ΣX; and the sample mean, M. 2. The school district decides it will be easier to understand the scores if they are converted to a 100-point scale. This requires that each score be multiplied by 4. Calculate the new n, ΣX, and M.
1. n = 12 ; SigmaX = 192 ; M = 16 2. n = 12 ; SigmaX = 768 ; M = 64
Computing the mean from a frequency distribution table 1. Calculate n, ΣX, and M. (Note: ΣX denotes the sum of all scores across the entire sample, not the sum of the column labeled X.) 2.The two patients who reported more than five exercise sessions during the reference week later admit that they were dishonest, so you decide to discard their data. Calculate the new n, ΣX, and M.
1. n = 25 ; SigmaX = 71 ; M = 2.84 2. n = 23 ; SigmaX = 58 ; M = 2.52
#6 Calculating the mean when adding or subtracting a constant 1. A professor gives a statistic exam. The exam has 25 possible points. The scores for the students in the first classroom are as follows: 24,25,19,15,20,19,17,25, 17. Calculate the sample size, n, and the sample mean, M. 2. While grading the exam, the professor realizes that one of the questions covered material that was not et covered in the lectures. This question was worth 2 points, so she decides to add 2 points to everyone's score.
1. n = 9 ; M = 20.11 2. n = 9 ; M = 22.11
#9 Finding the mode Consider the sample of n = 8 scores whose frequency distribution histogram is shown here. 1. The histogram is: 2. Place a purple diamond above the mode, or above each mode, if there is more than one. (Note: To receive credit, you must drag the diamond(s) into the center of the box directly above the mode(s) you want to mark.)
1. unimodal 2. Place the diamond on top of the box of the highest box. So for this problem mine was #2 bc it had the most boxes, which was 3 blocks high. Basically place it at the highest peak. Should be in a white box not the gray one.
#12 Selecting a measure of central tendency Suppose you collect a sample of 1,000 people. For each person in your sample, you have measured a value for each of the variables listed in the first column of the following table. For each variable, select which of the measures of central tendency are appropriate to use to describe your sample's values for that variable. Check all that apply.
Gender = Mode Height in inches = mean , median and mode Level of schooling completed = mode and median
#1 Engagement Activity : Central Tendency
Helpful: The older group had an average reaction time of 760ms ; The younger group had an average reaction time 930ms. Not so helpful: The younger group seemed to be faster than the older group ; Half of the older group had a faster than 800ms answer time.
#12 Explanation
Recall that the mode is the value of a variable that occurs the most frequently within the sample. The median is a value of a variable selected such that half of the values within the sample are above that value and half are below. Finally, the mean is the arithmetic average, that is, the sum of all the values of the variable divided by the number of values. Gender is measured on a nominal scale, so the only measure that can be calculated (of these three measures of central tendency) is the mode. Since a variable measured on a nominal scale does not consist of ordered categories, a median cannot be calculated. Also, since a variable measured on a nominal scale consists of names and not values, mathematical operations such as taking an average are not possible. Level of schooling completed (not a high school graduate/high school graduate/college graduate/postgraduate degree) is measured on an ordinal scale. As with variables measured on a nominal scale, the mode (or modes) can be determined because you can decide which value (or values) occurs with the most frequency. You can also determine a median because the categories are ordered so that you can determine a value such that half the values within the sample are above and half are below. When a variable measured on an ordinal scale consists of names and not values, mathematical operations such as taking an average are not possible. When a variable measured on an ordinal scale consists of numbers, mathematical operations such as taking an average are possible but given that numbers used in ordinal rankings are not necessarily equidistant from one another, mathematical operations such as taking an average are not appropriate. Finally, height in inches is measured on a ratio scale. Since variables measured on either an interval or a ratio scale are numerical, it is possible to calculate a mode, a median, and a mean.
#11 Explanation
The mean for a distribution is algebraically defined as the arithmetic average, that is, the sum of all of the scores divided by the number of scores. Since it is calculated using a well-defined equation, there is only one value for the mean, and that value may not correspond to an actual score in the data. This value, however, can be easily affected by extreme scores. It cannot be computed for an open-ended distribution, for data with undetermined values, or for data that are measured on a nominal scale. The median is the score that divides a distribution exactly in half. There is only one such score. The median, however, is not algebraically defined so there is no equation you use to calculate its value. It is not easily affected by extreme scores, as it does not matter how extreme the scores are; it only matters on which side of the median the scores lie. For similar reasons, the median can be found for an open-ended distribution or for data with undetermined values. In addition, the median is appropriate for ordinal data. The mode is the score or category that has the greatest frequency. The mode is particularly useful because it is the only measure of central tendency that necessarily corresponds to an actual score in the data and because it is the only measure of central tendency that can be used for data that are measured on a nominal scale. There can be more than one mode, as in a bimodal distribution. The mode, however, is not algebraically defined like the mean and does not correspond to a particular percentile, such as the 50th percentile.
#13 Central tendency and distribution shape
The right tail of the distribution of the incomes extends farther than the other tail. Therefore, the distribution is positively skewed. The mean is $71,200 , and the median is $70,600. The mean is greater than the median. The left tail of the distribution of the retirement ages extends farther than the other tail. Therefore, the distribution is negatively skewed. The mean is 62.9 years , and the median is 63.8 years. The mean is less than the median. When the distribution is symmetrical, the mean is equal to the median. When the distribution is positively skewed, the mean is usually greater than the median. When the distribution is negatively skewed, the mean is usually less than the median. The presence of extremely large or small values in the data affects the mean more than the median. Therefore the median is the preferred measure of central tendency when the distribution is skewed.
#8 Explanation
Whether the scores are measurements of a discrete or a continuous variable, the median divides the area of the tiles exactly in half. When the variable is continuous, however, boxes located in the same column may have different measurements. Boxes in the X = 3 column, for example, have measurements in the interval from 2.5 (the lower limit of the interval) to 3.5 (the upper limit of the interval). The X-axis on the histogram is a continuous number line with the midpoints of the class intervals aligned with the centers of the columns and the real class limits are the edges of the columns. The median is the X-axis intercept of your vertical line. Recall that this vertical line divides the total area of the boxes in half. In this case, the vertical line has an X-intercept of 2.5 + 0.25 = 2.75. The value 2.5 is the lower real limit for the score X = 3, and 1/4 = 0.25 is the fraction of the boxes at X = 3 that you needed to add to the boxes to the left of X = 3 in order to ensure that half of the total area (one box) is to the left of your vertical line. Thus, when the variable is continuous, the median is 2.75.
