Measures of Central Tendency,variability, and postion
12th Grade math class.
organize the data 78,67,95,85,84,86,82,75,85,94,91,57,78,77,88,90,91,86,84,82,75,58,73,75. Then do 4 columns of Grades,Deviations,Squares,and Z-Scores
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean for deviation. Square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
Z score = difference/standard deviation
I'll let you do the calculations.
To organize the data into four columns of Grades, Deviations, Squares, and Z-Scores, follow these steps:
Step 1: Calculate the Measures of Central Tendency
- The Measures of Central Tendency are used to describe the center or average of the data. Common measures include the mean, median, and mode.
- To find the mean, also known as the average, add up all the grades and divide by the total number of grades (in our case, 24).
- Sum of grades: 78 + 67 + 95 + 85 + 84 + 86 + 82 + 75 + 85 + 94 + 91 + 57 + 78 + 77 + 88 + 90 + 91 + 86 + 84 + 82 + 75 + 58 + 73 + 75 = 2041
- Mean: 2041 / 24 = 85.04
- To find the median, arrange the grades in numerical order and find the middle value. If there are two middle values, calculate their average.
- Arrange grades in ascending order: 57, 58, 67, 73, 75, 75, 75, 77, 78, 78, 82, 82, 84, 84, 85, 85, 86, 86, 88, 90, 91, 91, 94, 95
- Since there are 24 grades, the median is the 12th and 13th values: (82 + 84) / 2 = 83
- To find the mode, identify the grade value(s) that appear most frequently in the data set.
- In this case, there is no mode since no grade value repeats.
Step 2: Calculate the Measures of Variability
- The Measures of Variability provide information about the spread or dispersion of the data. Common measures include the range, variance, and standard deviation.
- To find the range, subtract the minimum grade from the maximum grade.
- Minimum grade: 57
- Maximum grade: 95
- Range: 95 - 57 = 38
- To find the variance, calculate the average of the squared deviations from the mean.
- Calculate deviations: For each grade, subtract the mean from the grade.
- Example: Deviation for the first grade (78) = 78 - 85.04 = -7.04
- Calculate squares: Square each deviation.
- Example: Square of -7.04 = (-7.04)^2 = 49.5616
- Calculate the variance: Sum the squared deviations and divide by the total number of grades.
- Sum of squares: Sum of all squared deviations
- Variance: Sum of squares / 24 (total number of grades)
- To find the standard deviation, take the square root of the variance.
Step 3: Calculate the Z-Scores
- Z-Scores indicate the number of standard deviations a grade is from the mean.
- To find the Z-Score for each grade, subtract the mean from the grade and divide by the standard deviation.
- Example for the first grade (78): (78 - 85.04) / (standard deviation)
Once you have performed these calculations, you can organize the data into the four columns: Grades, Deviations, Squares, and Z-Scores.