Find the interquartile range given these data points: 57,66,69,71,72,73,74,77,78,7879,79,81,81,82,83,83,88,89,94.
When I used the TI-84 calculator, I got ...
Q1=72.5
Q3=82.5
IQR=Q3-Q1=82.5-72.5=10
But..
When I created a stem and leaf plot I found that
Q2= (78+79)/2 =78.5
Q1=72
Q3=83
IQR= 83-72=11
So is it 10 or 11?
Did I do something wrong when using the stem and leaf plot?
There are 20 data in total.
Q1 should be (5th datum + 6th datum)/2
Q2 should be (10th datum + 11th datum)/2
Q3 should be (15th datum +16th datum)/2
Your own calculation is wrong.
Calculate yourself again.
(Ans: IQR=10)
You did not do anything wrong when using the stem and leaf plot. The discrepancy in the interquartile range (IQR) calculation arises from the fact that there are two different methods commonly used to calculate quartiles: the method employed by your TI-84 calculator and the method involving stem and leaf plots.
The TI-84 calculator uses the method known as Tukey's hinges, which provides a robust estimate of the quartiles by removing any outliers. According to this method, the first quartile (Q1) is calculated as the median of the lower half of the data, and the third quartile (Q3) is calculated as the median of the upper half of the data. Thus, Q1 is 72.5 and Q3 is 82.5, resulting in an IQR of 10.
However, when using a stem and leaf plot, the standard method considers the first quartile (Q1) as the median of the lower half of the data and the third quartile (Q3) as the median of the upper half of the data. In this case, Q1 is 72 and Q3 is 83, resulting in an IQR of 11.
Both methods are widely used and accepted, but they can sometimes yield slightly different results. The difference between the two methods is due to the way they treat odd and even sample sizes. The TI-84 calculator's method is more commonly used as it is less sensitive to outliers. Ultimately, the choice between the two methods depends on the context of your analysis and the specific requirements of your task.
To find the interquartile range (IQR) using the stem and leaf plot, you need to make sure you have accurately identified the quartiles.
In your case, you correctly identified Q1 as 72 and Q3 as 83. However, your calculation for Q2 seems to be incorrect. The median or Q2 is the value that divides the data into two equal halves.
From the stem and leaf plot, we can see that the 9th and 10th values are 78 and 79, respectively. To find the median, we need to take the average of these two values: (78 + 79) / 2 = 78.5.
So, your calculation for Q2 is correct: Q2 = 78.5.
Now, to find the IQR, you subtract Q1 from Q3: IQR = Q3 - Q1 = 83 - 72 = 11.
Therefore, the correct interquartile range for the given data points is 11.