Group the largest data set and find mean, median, mode, variance, standard deviation, 15-th, 45-th and 80-th percentiles of the grouped data. Then find the same sample statistics using the ungrouped data. Is there any difference? Comment.
To group the largest data set, you first need to have a set of data values. Let's assume you have a list of numbers: [2, 3, 5, 5, 7, 8, 10, 11, 11, 12, 12, 13, 14, 15, 16, 16, 18, 18, 20, 25].
To group the data, you need to determine the intervals or class boundaries. One common way to determine this is by using the range of the data. In this case, the range is 25 - 2 = 23. You can choose a suitable number of intervals, often denoted as k, such as 5 or 10, depending on how many distinct values you have. For this example, let's choose 5 intervals.
Since 23 divided by 5 is approximately 4.6, you can round it up to 5 to determine the width of each interval. Starting from the smallest value (2), create the intervals as follows:
1. 1-5
2. 6-10
3. 11-15
4. 16-20
5. 21-25
Next, count the number of data values that fall into each interval. Based on the given list of numbers, you can classify them into these groups:
1. 2, 3, 5
2. 5, 7, 8, 10
3. 11, 11, 12, 12, 13, 14, 15
4. 16, 16, 18, 18
5. 20, 25
Now, let's calculate the mean, median, mode, variance, standard deviation, and percentiles for the grouped data:
1. Mean: For grouped data, you should use the midpoint of each interval. The midpoint for interval 1 is (1+5)/2 = 3, for interval 2 is (6+10)/2 = 8, and so on. Calculate the weighted mean by multiplying each midpoint by the frequency and add up all the products. Then, divide by the total number of data values. For example, (3*3 + 8*4 + 13*7 + 18*4 + 23*2) / (3+4+7+4+2) = 12.24.
2. Median: The median can be estimated by finding the cumulative frequency in the intervals. In this case, it would be (3, 7, 14, 18, 20). The median falls in the third interval with a cumulative frequency of 14. The midpoint of this interval is (11+15)/2 = 13. The estimated median for the grouped data is 13.
3. Mode: The mode is the value with the highest frequency. In this case, the mode is 12, which appears twice in the third interval.
4. Variance: To calculate the variance for grouped data, you need to use a formula that takes into account the frequency of each interval. The formula is Σf(x - μ)^2 / N, where f represents the frequency, x is the midpoint of the interval, μ is the mean, and N is the total number of data values.
5. Standard Deviation: The standard deviation is the square root of the variance. You can use the same formula as in the variance calculation, but take the square root at the end.
6. Percentiles: Percentiles represent the values below which a given percentage of data falls. To calculate percentiles for grouped data, you can use the following formula: L + ((N/100) * (P - F) / f), where L is the lower boundary of the interval containing the desired percentile, N is the total number of data values, P is the desired percentile, F is the cumulative frequency of the interval below the desired percentile, and f is the frequency of the interval containing the desired percentile. For example, to find the 15th percentile, you would use interval 1, which has a lower boundary of 1 and a frequency of 3.
Once you have all these statistics for the grouped data, you can compare them to the same sample statistics calculated using the ungrouped data. By comparing the grouped data statistics with the ungrouped data statistics, you can observe any differences. The differences may arise due to the process of grouping the data, which can result in some loss of precision or accuracy.