Two sets of data are shown.
Data Set A: 30, 38, 42, 42, 43, 47, 51, 51, 57, 59
Data Set B: 38, 39, 40, 42, 44, 46, 47, 50, 51, 52
Question
Choose ,begin emphasis,all,end emphasis, the measures which are ,begin emphasis,greater,end emphasis, for data set A than for data set B.
The measures that are greater for data set A than for data set B are:
- Maximum value: The maximum value for data set A is 59, while for data set B it is 52. Therefore, the maximum value is greater for data set A.
- Median: The median for data set A is 47, while for data set B it is 46. Therefore, the median is greater for data set A.
- Third quartile: The third quartile for data set A is 57, while for data set B it is 50. Therefore, the third quartile is greater for data set A.
- Mean: The mean for data set A is approximately 47.7, while for data set B it is approximately 44.9. Therefore, the mean is greater for data set A.
To determine the measures that are greater for data set A than for data set B, we need to compare the values in both data sets.
Let's compare the measures one by one:
1. Median: The median is the middle value in a data set when it is arranged in ascending order. To find the median, we first need to rearrange the data sets in ascending order:
Data Set A: 30, 38, 42, 42, 43, 47, 51, 51, 57, 59
Data Set B: 38, 39, 40, 42, 44, 46, 47, 50, 51, 52
The medians for both data sets are as follows:
Median of Data Set A = 45
Median of Data Set B = 44
Since the median of Data Set A (45) is greater than the median of Data Set B (44), we can say that the median is greater for Data Set A than for Data Set B.
2. Mode: The mode is the value(s) that occur most frequently in a data set. In this case, both data sets have a mode of 42, as it appears twice in both sets.
The mode is the same for both data sets, so we cannot say that the mode is greater for Data Set A than for Data Set B.
3. Range: The range is the difference between the largest and smallest values in a data set. For Data Set A, the largest value is 59 and the smallest value is 30, giving a range of 59 - 30 = 29. For Data Set B, the largest value is 52 and the smallest value is 38, giving a range of 52 - 38 = 14.
Since the range of Data Set A (29) is greater than the range of Data Set B (14), we can say that the range is greater for Data Set A than for Data Set B.
In summary, the measures that are greater for Data Set A than for Data Set B are the median and the range. The mode is the same for both data sets.
To determine which measures are greater for Data Set A compared to Data Set B, we need to compare different statistical measures between the two sets. The commonly used measures of central tendency are mean, median, and mode. Additionally, we can also compare the range and interquartile range (IQR) to see if one set has a greater spread of values than the other.
1. Mean: Calculate the mean for both Data Set A and Data Set B by adding up all the values in each set and dividing by the number of values. In this case, the mean for Data Set A is (30 + 38 + 42 + 42 + 43 + 47 + 51 + 51 + 57 + 59) / 10 = 461 / 10 = 46.1, and the mean for Data Set B is (38 + 39 + 40 + 42 + 44 + 46 + 47 + 50 + 51 + 52) / 10 = 449 / 10 = 44.9. Therefore, the mean for Data Set A is greater than the mean for Data Set B.
2. Median: To find the median, arrange the values in each set in ascending order and determine the middle value. If there are two middle values, calculate their average. Data Set A is already sorted in ascending order, and since it contains an even number of values, the median is the average of the two middle values, which are 43 and 47. Therefore, the median is (43 + 47) / 2 = 45. For Data Set B, the values in the set also need to be sorted in ascending order. After sorting, the median is simply the middle value, which is 45. Therefore, the median is the same for both Data Set A and Data Set B.
3. Mode: The mode is the value(s) that appear most frequently in a set. In this case, both Data Set A and Data Set B do not have any repeated values, so there is no mode for either set.
4. Range: Calculate the range for each set by subtracting the minimum value from the maximum value. For Data Set A, the range is 59 - 30 = 29, and for Data Set B, the range is 52 - 38 = 14. Therefore, the range of Data Set A is greater than the range of Data Set B.
5. Interquartile Range (IQR): The IQR measures the spread of the middle 50% of the data. To calculate the IQR, first find the first quartile (Q1) and third quartile (Q3). Q1 is the median of the lower half of the data, and Q3 is the median of the upper half of the data. Then, calculate the difference between Q3 and Q1. Since both Data Set A and Data Set B have 10 values, we divide each set into two halves. For Data Set A, Q1 is the median of the first five values (38, 42, 42, 43, 47), which is 42. Q3 is the median of the last five values (51, 51, 57, 59), which is 54. Therefore, the IQR for Data Set A is 54 - 42 = 12. For Data Set B, Q1 is the median of the first five values (38, 39, 40, 42, 44), which is 40, and Q3 is the median of the last five values (46, 47, 50, 51, 52), which is 50. Hence, the IQR for Data Set B is 50 - 40 = 10. Therefore, the IQR for Data Set A is greater than the IQR for Data Set B.
In conclusion, the measures that are greater for Data Set A than for Data Set B are the mean and the range. The median, mode, and interquartile range are either the same or not applicable in this case.