The following data give the number of keyboards assembled at the Twentieth century electronics company for a sample of 25 days. 45, 52, 48,41,56,46,44,42,48,53,51,53,51,48,46,43,52,50,54,
47,44,47,50,49,52
a.calculate the values of the three quartiles and the interquartile range
b. Find the percentile rank of 50
Nice answer
a. To calculate the quartiles and the interquartile range, you need to arrange the data in ascending order:
41, 42, 43, 44, 44, 45, 46, 46, 47, 47, 48, 48, 48, 49, 50, 50, 51, 51, 52, 52, 52, 53, 53, 54, 56
1. First quartile (Q1): The median of the lower half of the data (the 25th percentile).
Q1 = (n+1)/4 = (25+1)/4 = 6th value = 45
2. Second quartile (Q2): The median of the entire data set (the 50th percentile).
Q2 = (n+1)/2 = (25+1)/2 = 13th value = 48
3. Third quartile (Q3): The median of the upper half of the data (the 75th percentile).
Q3 = (3n+1)/4 = (3*25+1)/4 = 19th value = 52
4. Interquartile range (IQR): The difference between Q3 and Q1.
IQR = Q3 - Q1 = 52 - 45 = 7
b. To find the percentile rank of 50, you can use the following formula:
Percentile Rank = (Number of values less than the given value + 0.5 * Number of values equal to the given value) / Total number of values
In this case, the number of values less than 50 is 13, and there are no values equal to 50. The total number of values is 25.
Percentile Rank = (13 + 0.5 * 0) / 25 = 13/25 = 0.52
Therefore, the percentile rank of 50 is 0.52, or 52%.
a. To calculate the quartiles and the interquartile range, you need to follow these steps:
Step 1: Sort the data in ascending order:
41, 42, 43, 44, 44, 45, 46, 46, 47, 47, 48, 48, 48, 49, 50, 50, 51, 51, 52, 52, 52, 53, 53, 54, 56
Step 2: Find the median, which is the second quartile (Q2):
The data has an odd number of values (25), so the middle value is the median.
Median: 48
Step 3: Find Q1, the first quartile:
To find Q1, first find the median of the lower half of the data.
Lower half: 41, 42, 43, 44, 44, 45, 46, 46, 47
Number of values: 9 (odd)
Median of lower half: (44 + 45) / 2 = 44.5
Q1: 44.5
Step 4: Find Q3, the third quartile:
To find Q3, first find the median of the upper half of the data.
Upper half: 50, 50, 51, 51, 52, 52, 52, 53, 53, 54, 56
Number of values: 11 (odd)
Median of upper half: 52
Q3: 52
Step 5: Calculate the interquartile range (IQR):
IQR = Q3 - Q1
IQR = 52 - 44.5
IQR = 7.5
Therefore, the values of the three quartiles are:
Q1: 44.5
Q2 (median): 48
Q3: 52
And the interquartile range is 7.5.
b. To find the percentile rank of 50, you can use the following formula:
Percentile Rank = (Number of values below 50 / Total number of values) * 100
Step 1: Count the number of values below 50, which is 13.
Step 2: Calculate the percentile rank:
Percentile Rank = (13 / 25) * 100 = 52
So, the percentile rank of 50 is 52.