Consider the following data set:
27,49,15,43,35,57,13,73,23,31,32,41,61, 39,7.
Suppose that you graph this data, with bins of size 10, with the first bin starting 1 below the smallest value in your data.
What is the range approximation for the standard deviation of this data?
What is the height of the fourth bar from the left in a relative frequency histogram?
What proportion of the data lies within two standard deviations of the mean?
To find the range approximation for the standard deviation, we need to calculate the standard deviation first. Here are the steps:
Step 1: Find the mean (average) of the data set.
27 + 49 + 15 + 43 + 35 + 57 + 13 + 73 + 23 + 31 + 32 + 41 + 61 + 39 + 7 = 559
Mean = 559 / 15 ≈ 37.27
Step 2: Find the difference between each data point and the mean, square each difference, and sum them up.
(27 - 37.27)^2 + (49 - 37.27)^2 + (15 - 37.27)^2 + (43 - 37.27)^2 + (35 - 37.27)^2 + (57 - 37.27)^2 + (13 - 37.27)^2 + (73 - 37.27)^2 + (23 - 37.27)^2 + (31 - 37.27)^2 + (32 - 37.27)^2 + (41 - 37.27)^2 + (61 - 37.27)^2 + (39 - 37.27)^2 + (7 - 37.27)^2 = 3083.2
Step 3: Divide the sum from Step 2 by the number of data points minus 1 (n-1) to find the variance.
Variance = 3083.2 / (15 - 1) ≈ 205.5467
Step 4: Take the square root of the variance to find the standard deviation.
Standard Deviation ≈ √205.5467 ≈ 14.3316
Therefore, the range approximation for the standard deviation of this data is approximately 14.3316.
To find the height of the fourth bar from the left in a relative frequency histogram, we need to calculate the relative frequency for each bin. Here are the steps:
Step 1: Organize the data into bins:
Bin 1: 0-9, Bin 2: 10-19, Bin 3: 20-29, Bin 4: 30-39, Bin 5: 40-49, Bin 6: 50-59, Bin 7: 60-69, Bin 8: 70-79.
Step 2: Count the occurrences of data points in each bin:
Bin 1: 2, Bin 2: 1, Bin 3: 2, Bin 4: 3, Bin 5: 3, Bin 6: 2, Bin 7: 2, Bin 8: 0.
Step 3: Calculate the relative frequency for each bin by dividing the count by the total number of data points (15):
Bin 1: 2 / 15 ≈ 0.1333, Bin 2: 1 / 15 ≈ 0.0667, Bin 3: 2 / 15 ≈ 0.1333, Bin 4: 3 / 15 ≈ 0.2, Bin 5: 3 / 15 ≈ 0.2, Bin 6: 2 / 15 ≈ 0.1333, Bin 7: 2 / 15 ≈ 0.1333, Bin 8: 0 / 15 = 0.
The height of the fourth bar from the left in a relative frequency histogram is approximately 0.2.
To find the proportion of the data that lies within two standard deviations of the mean, we need to calculate the boundaries for the range.
Step 1: Multiply the standard deviation by 2 and add/subtract it from the mean:
Upper boundary = 37.27 + (2 * 14.3316) ≈ 66.9332
Lower boundary = 37.27 - (2 * 14.3316) ≈ 7.6068
Step 2: Count the number of data points that lie within the upper and lower boundaries:
Number of data points within two standard deviations = 4 (43, 35, 57, 39)
Step 3: Calculate the proportion by dividing the number of data points within the boundaries by the total number of data points:
Proportion = 4 / 15 ≈ 0.2667
Therefore, approximately 26.67% of the data lies within two standard deviations of the mean.
To find the range approximation for the standard deviation of the data, you'll need to go through a few steps:
Step 1: Find the mean of the data set by summing up all the values and dividing by the number of values. In this case, the mean is (27 + 49 + 15 + 43 + 35 + 57 + 13 + 73 + 23 + 31 + 32 + 41 + 61 + 39 + 7) / 15 = 36.
Step 2: Calculate the deviation for each value by subtracting the mean from each value. In this case, the deviations are (-9, 13, -21, 7, -1, 21, -23, 37, -13, -5, -4, 5, 25, 3, -29).
Step 3: Square each deviation to eliminate any negative values and to emphasize larger deviations. In this case, the squared deviations are (81, 169, 441, 49, 1, 441, 529, 1369, 169, 25, 16, 25, 625, 9, 841).
Step 4: Find the sum of the squared deviations. In this case, the sum of squared deviations is 4202.
Step 5: Divide the sum of squared deviations by the number of values minus one, and then take the square root of that result to get the standard deviation. In this case, the standard deviation is sqrt(4202 / (15 - 1)) = sqrt(280.13) ≈ 16.73.
Therefore, the range approximation for the standard deviation of this data is approximately 16.73.
Now, let's move on to the height of the fourth bar from the left in a relative frequency histogram.
To find the height of the fourth bar, we first need to determine the number of data points that fall within the range represented by that bar. Since the bins have a size of 10, and the first bin starts 1 below the smallest value in the data, the fourth bar represents the range from 31 to 40.
Looking at the data set, the values that fall within this range are 31, 32, and 39. There are 3 data points in this range.
Next, we need to calculate the relative frequency, which is the proportion of data points that fall within the range of the fourth bar. To do this, we divide the number of data points in the range by the total number of data points in the data set. In this case, the relative frequency is 3 / 15 = 0.2.
Therefore, the height of the fourth bar from the left in the relative frequency histogram is 0.2.
Finally, let's determine the proportion of the data that lies within two standard deviations of the mean.
To do this, we need to calculate the interval within which two standard deviations from the mean fall. Since the standard deviation is approximately 16.73, we multiply it by 2 to get 2 * 16.73 = 33.46.
The interval would be from the mean minus 33.46 to the mean plus 33.46. Since the mean is 36, the interval is from 36 - 33.46 to 36 + 33.46, which gives us the range from 2.54 to 68.46.
Now we need to count the number of data points that fall within this range. Looking at the data set again, the values that fall within this range are 27, 49, 15, 43, 35, 57, 13, 23, 31, 32, 41, 61, and 39. There are 13 data points in this range.
Finally, we calculate the proportion by dividing the number of data points in the range by the total number of data points in the data set. In this case, the proportion is 13 / 15 = 0.87, or 87%.
Therefore, approximately 87% of the data lies within two standard deviations of the mean.