Question

self-check 2.5, 2.7

1. the following data show the number of months patients typically wait on a transplant list before getting surgery. the data are ordered from smallest to largest.

3; 4; 5; 7; 7; 7; 7; 8; 8; 9; 9; 10; 10; 10; 10; 10; 11; 12; 12; 13; 14; 14; 15; 15; 17; 17; 18; 19; 19; 19; 21; 21; 22; 22; 23; 24; 24; 24; 24
a) calculate the mean.

b) identify the median.

c) identify the mode.

d) calculate the standard deviation.

e) calculate the variance.

Explanation:

Step1: Count total data points

Count all values: $n = 40$

Step2: Calculate sum of all values

$$3+4+5+7+7+7+7+8+8+9+9+10+10+10+10+10+11+12+12+13+14+14+15+15+17+17+18+19+19+19+21+21+22+22+23+24+24+24+24 = 539$$

Step3: Compute the mean (part a)

Mean formula: $\bar{x} = \frac{\sum x}{n}$
$\bar{x} = \frac{539}{40} = 13.475$

Step4: Find the median (part b)

For even $n$, median = average of $\frac{n}{2}$ and $\frac{n}{2}+1$ values:
$\frac{40}{2}=20$, $\frac{40}{2}+1=21$. The 20th value is 13, 21st is 14.
Median = $\frac{13+14}{2} = 13.5$

Step5: Identify the mode (part c)

Find most frequent value: 10 and 24 each appear 5 times.

Step6: Compute squared deviations (for variance)

Calculate $\sum (x-\bar{x})^2$:
$$(3-13.475)^2 + (4-13.475)^2 + (5-13.475)^2 + 4(7-13.475)^2 + 2(8-13.475)^2 + 2(9-13.475)^2 + 5(10-13.475)^2 + (11-13.475)^2 + 2(12-13.475)^2 + (13-13.475)^2 + 2(14-13.475)^2 + 2(15-13.475)^2 + 2(17-13.475)^2 + (18-13.475)^2 + 3(19-13.475)^2 + 2(21-13.475)^2 + 2(22-13.475)^2 + (23-13.475)^2 + 4(24-13.475)^2$$
$$= 109.7256 + 89.7756 + 71.8256 + 4(41.9256) + 2(29.9756) + 2(19.9256) + 5(12.0756) + 6.1256 + 2(2.1756) + 0.2256 + 2(0.2756) + 2(2.2756) + 2(12.6256) + 20.8756 + 3(30.8756) + 2(56.6256) + 2(72.6256) + 90.7256 + 4(111.6006)$$
$$= 109.7256+89.7756+71.8256+167.7024+59.9512+39.8512+60.378+6.1256+4.3512+0.2256+0.5512+4.5512+25.2512+20.8756+92.6268+113.2512+145.2512+90.7256+446.4024 = 1589.175$$

Step7: Calculate variance (part e)

Sample variance formula: $s^2 = \frac{\sum (x-\bar{x})^2}{n-1}$
$s^2 = \frac{1589.175}{39} \approx 40.748$

Step8: Calculate standard deviation (part d)

Standard deviation = square root of variance: $s = \sqrt{s^2}$
$s = \sqrt{40.748} \approx 6.383$

Answer:

a) 13.475
b) 13.5
c) 10 and 24
d) 6.383
e) 40.748