Boxplot and percentile activity. The following are the pulse rate data (bpm) rep

ID: 3225925 • Letter: B

Question

Boxplot and percentile activity. The following are the pulse rate data (bpm) reported for in the first day survey (Spr 2016). 45 58 60 60 64 65 6? 68 68 68 69 69 70 72 73 74 76 79 80 81 84 90 100 For these data calculate the mean, median, variance, and standard deviation. What value is associated with the 25^th percentile? The 75^th percentile? The 10^th percentile? Draw a modified boxplot for these data (horizontal). Make sure to label your axis! What is the shape of this distribution? Is it symmetric or skewed? If skewed, is it right or left- Are there any outliers indicated by the boxplot? If there are, list their value(s) Are pulse rates discrete or continuous? Why? Interval or ratio measurement level? Why? Say you have a pulse rate of 80 bpm. What percentile is that pulse rate be in these data?

Explanation / Answer

Given, n =23, x2 = 119816, (x)2 = 2689600.

Also, the values (observations) are given in ascending order => it is an ordered set.

Part (a)

Mean = (x)/n = {sqrt(2689600)}/23 = 71.3043 ANSWER

Part (b)

Median is the middle value in the ordered set. Since n = 23,

the middle value = 12th value = 69 ANSWER

Part (c)

Variance = {(x2)/n} – {(x)2/n2} = {(119816)/23 – {(2689600/232} = 125.0813. ANSWER

Part (d)

Standard deviation = sqrt{variance} = 125.0813 = 11.184 ANSWER

Part (e)

To find 25th percentile:

Back-up Theory

For 0 < p < 1, pth quantile of a set of observations is that observation in the

ordered set such that p-fraction of the observations lie below and

(1 - p) fraction of the observations lie above that observation.

Empirically, find i = np + 0.5. Let the ordered set be x(1), x(2), x(3), ……, x(n).

If i is an integer, then, x(i) is the pth quantile. If i is not an integer, find the

integer just lower than i, say j, Then, interpolate between x(j) and (j + 1).

To find 25th percentile, p = 0.25 and given n=23

I = 6.25.

Since 6.25 is not an integer, j = 6 and 25th percentile is interpolated between 6th and 7th value of the ordered set

6th value = 65 and 7th value = 67. So, 6.25th value =

65 + (67 – 65)x0.25 = 65.5 ANSWER

similarly, 75th percentile = 79.25

10th percentile = 59.16