Question

1. the data set gives the numbers of homeruns for the 10 batters who hit the most homeruns during the 2005 major league baseball regular season. 51, 48, 47, 41, 45, 43, 46, 40, 39, 40

mean median mode range standard deviation
min q1 median q3 max
\okay, i have the box. but, i need your help to complete my box - and - whisker plot.\ (with a cartoon of a dog and a cat, the dog is holding a box, the cat is thinking \i dont like where this is going.\)

Explanation:

Step1: Organize the data

First, sort the data set: \( 39, 40, 41, 43, 45, 46, 47, 48, 51 \) (Wait, there are 9 numbers? Wait, original data: 51, 48, 47, 41, 45, 43, 46, 40, 39, 40. Oh, I missed a 40. So sorted data: \( 39, 40, 40, 41, 43, 45, 46, 47, 48, 51 \) (n=10).

Step2: Calculate Mean

Sum the data: \( 39 + 40 + 40 + 41 + 43 + 45 + 46 + 47 + 48 + 51 \)
\( 39+40=79; 79+40=119; 119+41=160; 160+43=203; 203+45=248; 248+46=294; 294+47=341; 341+48=389; 389+51=440 \)
Mean \( \bar{x} = \frac{440}{10} = 44 \)

Step3: Calculate Median

For n=10 (even), median is average of 5th and 6th terms. 5th term: 43, 6th term: 45. Median \( = \frac{43 + 45}{2} = 44 \)

Step4: Find Mode

Mode is the most frequent number. 40 appears twice, others once. So Mode = 40

Step5: Calculate Range

Range = Max - Min = 51 - 39 = 12

Step6: Calculate Standard Deviation

First, find deviations from mean (\( x - \bar{x} \)):
39-44=-5; 40-44=-4; 40-44=-4; 41-44=-3; 43-44=-1; 45-44=1; 46-44=2; 47-44=3; 48-44=4; 51-44=7
Square the deviations:
25, 16, 16, 9, 1, 1, 4, 9, 16, 49
Sum of squared deviations: \( 25 + 16 + 16 + 9 + 1 + 1 + 4 + 9 + 16 + 49 = 146 \)
Variance \( s^2 = \frac{146}{10 - 1} \approx 16.222 \) (since sample standard deviation, divide by n-1? Wait, the problem doesn't specify population or sample. If it's population (all 10 batters), divide by n=10: \( \frac{146}{10} = 14.6 \), then standard deviation \( s = \sqrt{14.6} \approx 3.82 \). Wait, let's check again. Wait, sorted data: 39,40,40,41,43,45,46,47,48,51. Wait, 40 appears twice, correct. Sum of (x - x̄)^2:
(39-44)²=25, (40-44)²=16 (twice: 32), (41-44)²=9, (43-44)²=1, (45-44)²=1, (46-44)²=4, (47-44)²=9, (48-44)²=16, (51-44)²=49. So 25 + 32 + 9 + 1 + 1 + 4 + 9 + 16 + 49 = 25+32=57; 57+9=66; 66+1=67; 67+1=68; 68+4=72; 72+9=81; 81+16=97; 97+49=146. Correct. If population (σ), then σ²=146/10=14.6, σ=√14.6≈3.82. If sample (s), s²=146/9≈16.22, s≈4.03. But since it's all 10 batters (the entire population of top 10), use population standard deviation: √14.6≈3.82.

Step7: Box-and-Whisker Plot Values

Min = 39
Q1 (25th percentile): For n=10, Q1 is the median of first 5 terms: 39,40,40,41,43. Median of these is 40.
Median (50th percentile) we found as 44.
Q3 (75th percentile): Median of last 5 terms: 45,46,47,48,51. Median is 47.
Max = 51

Answer:

(Filling the table):

Mean	Median	Mode	Range	Standard Deviation

Min	Q1	Median	Q3	Max

(Note: Standard deviation calculation: if using population, √(146/10)=√14.6≈3.82; if sample, √(146/9)≈4.03. Depending on context, but since it's all 10, population is more appropriate here.)