The following is a set of data from a sample of n = 5:
7 4 9 8 The following is a set of data from a sample of n = 5:
7 4 9 8 2
a. Compute the mean, median, and mode.
b. Compute the range, variance, standard deviation, and coefficient of variation.
c. Compute the Z scores. Are there any outliers?
d. Describe the shape of the data set.
The first set of data only has n = 4.
Mode is the most frequently occurring score, if any.
Median = 50th percentile.
Range = highest score - lowest
The mean = sum of scores/number of scores
Subtract each of the scores from the mean and 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-mean)/SD
I'll let you do the calculations.
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a. To compute the mean, median, and mode for the given dataset:
Mean:
To calculate the mean, you need to sum up all the numbers in the dataset and divide it by the total count of the numbers.
7 + 4+ 9 + 8 + 2 = 30
Mean = 30 / 5 = 6
Median:
To calculate the median, you need to arrange the numbers in ascending order and then identify the middle value. If there are even numbers, take the average of the two middle values.
2, 4, 7, 8, 9
Since there is an odd number of values, the middle value is 7, so the median is 7.
Mode:
The mode is the value that appears most frequently in the dataset. In this case, all the numbers occur only once, so there is no mode.
b. To compute the range, variance, standard deviation, and coefficient of variation:
Range:
The range is calculated by subtracting the smallest value from the largest value.
Largest value: 9
Smallest value: 2
Range = 9 - 2 = 7
Variance:
To calculate the variance, you need to find the average of the squared differences from the mean.
(7-6)^2 + (4-6)^2 + (9-6)^2 + (8-6)^2 + (2-6)^2 = 34
Variance = 34 / 5 = 6.8
Standard Deviation:
The standard deviation is the square root of the variance.
Standard Deviation = √6.8 = 2.61
Coefficient of Variation:
The coefficient of variation measures the relative variation in the dataset, which is the standard deviation divided by the mean.
Coefficient of Variation = (2.61 / 6) × 100 = 43.5%
c. To compute the Z-scores:
The Z-score represents the number of standard deviations an individual data point is from the mean.
Z-score = (X - Mean) / Standard Deviation
For each value in the dataset, calculate the Z-score:
Z-score for 7 = (7 - 6) / 2.61 = 0.383
Z-score for 4 = (4 - 6) / 2.61 = -0.766
Z-score for 9 = (9 - 6) / 2.61 = 1.15
Z-score for 8 = (8 - 6) / 2.61 = 0.766
Z-score for 2 = (2 - 6) / 2.61 = -1.53
By analyzing the Z-scores, we can see that there are no outliers since all the values fall within a reasonable range, typically within ±3 standard deviations.
d. To describe the shape of the dataset:
Based on the given data, it's difficult to determine the shape accurately without additional data points. However, with only five data points, we may not be able to make any definitive conclusions about the shape of the dataset.
a. To compute the mean, add up all the numbers in the data set and divide by the total number of data points (5 in this case). So, for the first set of data, the mean is: (7 + 4 + 9 + 8 + 2) / 5 = 6.
To find the median, arrange the data set in ascending order and find the middle value. If there are an odd number of data points, the median is the middle value. If there are an even number of data points, the median is the average of the two middle values. So, for the first set of data, the median is 7.
The mode is the value that appears most frequently in the data set. For the first set of data, there is no mode as each number appears only once.
b. The range is the difference between the largest and smallest values in the data set. For the first set of data, the range is: 9 - 2 = 7.
To calculate the variance, subtract the mean from each data point, square the result, sum up all the squared differences, and divide by the total number of data points minus 1. So, for the first set of data, the variance is: [(7 - 6)^2 + (4 - 6)^2 + (9 - 6)^2 + (8 - 6)^2 + (2 - 6)^2] / (5 - 1) = 17.5.
The standard deviation is the square root of the variance. For the first set of data, the standard deviation is: √17.5 ≈ 4.18.
The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage. For the first set of data, the coefficient of variation is: (4.18 / 6) * 100 ≈ 69.67%.
c. To compute the Z scores, subtract the mean from each data point and divide the difference by the standard deviation. The Z scores indicate how many standard deviations a data point is from the mean. To identify outliers, typically a Z score greater than 3 or less than -3 is considered to be an outlier. For the first set of data, the Z scores are: (7 - 6) / 4.18 ≈ 0.24, (4 - 6) / 4.18 ≈ -0.48, (9 - 6) / 4.18 ≈ 0.72, (8 - 6) / 4.18 ≈ 0.48, (2 - 6) / 4.18 ≈ -0.96. None of the Z scores exceed 3 or fall below -3, so there are no outliers in this data set.
d. To describe the shape of the data set, one can observe the values and see if it follows a specific pattern. In this case, the data set appears to be relatively symmetric with no obvious outliers. However, since the data set is small, it's difficult to make definitive conclusions about the shape without further analysis or comparing it to a known distribution.