3. Indicate whether the given statement could apply to a data set consisting of 1,000 values that are all different.
a. The 29th percentile is greater than the 30th percentile.
b. The median is greater than the first quartile.
c. The third quartile is greater than the first quartile.
d. The mean is equal to the median.
e. The range is zero.
To answer whether the given statements could apply to a data set consisting of 1,000 values that are all different, we'll need to understand what each statement means and then analyze if it is possible or not.
a. The 29th percentile is greater than the 30th percentile:
To determine percentiles, we arrange the values in order from smallest to largest and find the position of a given percentile. In a data set of 1,000 different values, both the 29th percentile and the 30th percentile correspond to individual values in the data set. Therefore, it is not possible for the 29th percentile to be greater than the 30th percentile.
b. The median is greater than the first quartile:
The median is the middle value in a sorted data set. In a data set of 1,000 different values, the median will correspond to the 500th value. The first quartile represents the 25th percentile and corresponds to the position of the 250th value. Since the median is larger than the first quartile, it is possible for this statement to apply.
c. The third quartile is greater than the first quartile:
Similar to the previous statement, the first quartile represents the 25th percentile, and the third quartile represents the 75th percentile. In a data set of 1,000 different values, it is possible for the value at the 75th percentile to be greater than the value at the 25th percentile. Therefore, this statement could apply.
d. The mean is equal to the median:
To find the mean, we sum all the values in the data set and divide by the total number of values. In a data set of 1,000 different values, where each value is unique, it is highly unlikely that the sum of all values will be divisible by 1,000 without leaving a remainder. As a result, the mean is unlikely to equal the median.
e. The range is zero:
The range is the difference between the maximum and minimum values in a data set. In a data set of 1,000 different values, all values are unique, so the maximum value will be greater than the minimum value. Therefore, the range cannot be zero.
In summary:
a. This statement could not apply.
b. This statement could apply.
c. This statement could apply.
d. This statement is unlikely to apply.
e. This statement could not apply.
b and c are definitely true. Median = second quartile.
d can be true, but is not necessarily so.
1, 2, 3, 4, 5, 6, 7, both are 4
1, 2, 3, 4, 5, 6, 10, median still 4, but mean is higher. Mean is most effected by deviant scores.
a and e are false.