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Mean: 84.0625
Median: 87
Mode: 92
Range: 64
Standard Deviation: Approximately 14.95
Outliers: 36
We want to find the mean, median, mode, range, and standard deviation of the given data set, as well as whether there are any outliers.
The mean of a data set is the sum of the values divided by the total number of values in the set. Let's start by calculating the sum of the given values. 85+100+92+36+74+88+92+86+ 88+82+98+70+78+92+84+100 = 1345 There are 16 values in our set, so we have to divide the sum by 16. Mean: 1345/16=84.0625
Median: 86+ 88/2=87
The mode is the value or values that appear most often in a set of data. Let's find the mode of the given values. 85, 100, 92, 36, 74, 88, 92, 86 88, 82, 98, 70, 78, 92, 84, 100 The value that appears most often is 92. Mode: 92
The range is the difference between the greatest and least values in a set of data. For our exercise, the greatest value is 100 and the least value is 36. Range: 100-36=64
The standard deviation of a set of data is the average amount by which each individual value deviates or differs from the mean. Standard Deviation sqrt((x_1-μ )^2+(x_2-μ )^2+... +(x_n-μ )^2/n) In the above formula, x_1, ... ,x_n are the values of the set of data, μ is the mean, and n is the number of values. For this exercise, the mean is the sum of the values 1345 divided by the number of values 16. μ= 1345 16=84.0625 Let's use this value and apply the formula to each value in the set.
| x_n | x_n-μ | (x_n-μ)^2 |
|---|---|---|
| 85 | 85-84.0625=0.9375 | 0.9375^2≈ 0.9 |
| 100 | 100-84.0625=15.9375 | 15.9375^2≈ 254 |
| 92 | 92-84.0625=7.9375 | 7.9375^2≈ 63 |
| 36 | 36-84.0625=- 48.0625 | (- 48.0625)^2≈ 2310 |
| 74 | 74-84.0625=- 10.0625 | (- 10.0625)^2≈ 101.3 |
| 88 | 88-84.0625=3.9375 | 3.9375^2≈ 15.5 |
| 92 | 92-84.0625=7.9375 | 7.9375^2≈ 63 |
| 86 | 86-84.0625=1.9375 | 1.9375^2≈ 3.8 |
| 88 | 88-84.0625=3.9375 | 3.9375^2≈ 15.5 |
| 82 | 82-84.0625=- 2.0625 | (- 2.0625)^2≈ 4.3 |
| 98 | 98-84.0625=13.9375 | 83.9375^2≈ 194.3 |
| 70 | 70-84.0625=- 14.0625 | (- 14.0625)^2≈ 197.8 |
| 78 | 78-84.0625=- 6.0625 | (- 6.0625)^2≈ 36.8 |
| 92 | 92-84.0625=7.9375 | 7.9375^2≈ 63 |
| 84 | 84-84.0625=- 0.0625 | (- 0.0625)^2≈ 0.004 |
| 100 | 100-84.0625=15.9375 | 15.9375^2≈ 254 |
| Sum of Values | ≈ 3577.204 | |
Finally, we need to divide by 16 and then calculate the square root. Standard Deviation: sqrt(3577.204/16)≈ 14.95
To find the outliers, let's start by defining some important characteristics of data sets.
Let's find the upper quartile, the lower quartile, and the interquartile range for the given data set. Do not forget to write the values in numerical order! 36, 70, 74, 78, 82, 84, 85, 86 | 88, 88, 92, 92, 92, 98, 100, 100 We have two middle values for each half. Thus, we need to calculate the mean of those middle values. Upper Quartile:& 92+ 922= 92 Lower Quartile:& 78+ 822= 80 Interquartile Range:& 92- 80= 12 An outlier is an extremely high or extremely low value when compared with the rest of the values in the set. To check for them, we look for data values that are beyond the upper or lower quartiles by more than 1.5 times the interquartile range. Minimum=& Q_1-IQR*1.5 Maximum=& Q_2+IQR*1.5 Let's find the minimum and maximum for a value not to be an outlier. ccc Minimum& &Maximum 80-1.5* 12=62 & & 92+1.5* 12=110 Any value between 62 and 110 is not an outlier. Thus, the data set has only one outlier, 36.