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Flashcards · 0479 · Topic 5

Quartiles, percentiles and measures of dispersion flashcards

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Topic 5: Quartiles, percentiles and measures of dispersion

What are the lower and upper quartiles?
Lower quartile Q1: the median of the lower half of the ordered data, the 25th percentile. Upper quartile Q3: the median of the upper half, the 75th percentile.
What is the k-th percentile?
The value with k% of the data at or below it.
What is the difference between the range and the interquartile range?
Range = largest − smallest, so it uses only the two extremes. Interquartile range = Q3 − Q1, the spread of the middle half of the data.
Find the quartiles and IQR of 23, 31, 18, 27, 35, 29, 21
Ordered: 18, 21, 23, 27, 29, 31, 35. Median 27, left out of both halves. Q1 = 21, Q3 = 31, so IQR = 31 − 21 = 10.
At which cumulative frequencies do you read the quartiles and percentiles from a graph?
Q1 at n/4, median at n/2, Q3 at 3n/4, and the k-th percentile at (k/100) × n. Read across from the vertical axis to the curve, then down.
How do you find which percentile a value is from a cumulative frequency graph?
Go up from the value to the curve and across to read the cumulative frequency, then divide by n and multiply by 100. Example: 63 out of 90 → 70th percentile.
What are the two formulae for the standard deviation of n values?
Form 1: √[Σ(x − x̄)² ÷ n] Form 2: √[Σx² ÷ n − (Σx ÷ n)²], the mean of the squares minus the square of the mean.
What is the standard deviation formula for a frequency table?
√[Σfx² ÷ Σf − (Σfx ÷ Σf)²]. For grouped data use the class midpoints as x, and state that the answer is an estimate.
What is the link between the variance and the standard deviation?
The variance is the quantity under the square root; the standard deviation is the square root of the variance. A common slip is giving the variance and forgetting the square root.
Find the standard deviation of 7, 3, 9, 5, 6
Mean = 30 ÷ 5 = 6. Squared deviations 1, 9, 9, 1, 0 add to 20. SD = √(20 ÷ 5) = √4 = 2.
Which measure of spread is least affected by one extreme value?
The interquartile range. One extreme value changes the range and the standard deviation a lot, but the IQR may not change at all.
A grouped table has an open-ended top class. Which measures of spread can you still find?
The range and the standard deviation cannot be calculated. The IQR can still be estimated if both quartiles lie outside the open-ended class.

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