PC4.6: Statistics
Memorize:
1
𝑠 = √𝑛 ∑𝑛𝑖=1(𝑥𝑖 − 𝑥)2
𝐼𝑄𝑅 = 𝑄3 − 𝑄1
68%, 95%, 99.7%
Vocab: mean (𝑥), median, mode, bimodal, multimodal, range, inner quartile range (IQR), five number
summary (min, Q1, med, Q3, max), outlier, standard deviation (s), variance (𝑠 2 ), normal distribution
Warm-up: Find the mean, median, mode, and range for the following data:
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Mean:
Number of Satellites
0
0
1
2
28
30
21
8
1
0+0+1+2+28+30+21+8+1
9
=
91
9
≈ 10.1, Median: 2, Mode(s): {0,1}, Range: 30
Problems: Solve each problem:
In 1998, a total of 116,517 students took the AP Calc AB exam. Calculate the mean, median, mode, and
range of the AP Exam Grades from the table.
AP Exam Grade
5
4
3
2
1
Mean:
Number of students
18522
27102
31286
20732
18875
(5∙18522)+(4∙27102)+(3∙31286)+(2∙20732)+(1∙18875)
116517
Median: There are 116517 entries, pick the
Mode: The most frequent grade is 3
Range: 5 − 1 = 4
116517+1
2
355215
= 116517 ≈ 3.05
= 58259𝑡ℎ: 3
Using Mickey Mantle’s table of home runs per year, state the 5 number summary and IQR. Create a boxand-whiskers plot of the data. Determine which data points, if any, are outliers. Find the standard
deviation and variance.
Year
1951
1952
1953
1954
1955
1956
1957
1958
1959
Home Runs
13
23
21
27
37
52
34
42
31
Min Q1 Med Q3
13 21 28.5 37
IQR: 𝑄3 − 𝑄1 = 16
10
Year
1960
1961
1962
1963
1964
1965
1966
1967
1968
Home Runs
40
54
30
15
35
19
23
22
18
Max
54
15
20
25
30
35
40
45
50
55
Outliers exist beyond 𝐼𝑄𝑅 ∗ 1.5 = 16 ∗ 1.5 = 24 units of the IQR: less than -3 and greater than 61,
∴ ∄ Outliers
𝑠 = 11.9, 𝑠 2 = 142.7
Using Roger Maris’ table of home runs per year, state the 5 number summary and IQR. Create a boxand-whiskers plot of the data. Determine which data points, if any, are outliers. Find the standard
deviation and variance.
Year
1957
1958
1959
1960
1961
1962
1963
Home Runs
Year
14
56
16
39
68
33
23
1964
1965
1966
1967
1968
1969
Min Q1 Med Q3
8
12.5 23 47.5
IQR: 𝑄3 − 𝑄1 = 35
5
10
15
Home Runs
26
8
13
9
12
83
Max
83
20
25
30
35
40
45
50
55
60
Outliers exist beyond 𝐼𝑄𝑅 ∗ 1.5 = 25 ∗ 1.5 = 37.5 units of the IQR: less than -26.5 and greater than
73.5, ∴ ∄ Outliers
𝑠 = 24.3, 𝑠 2 = 588.9
Mr. Wytiaz gives a pre-calculus exam. The scores are normally distributed with a mean of 85 and a
standard deviation of 3. What percent of the students would you expect to have a score that is: between
82% and 88%? between 88% and 91%? between 79 and 91%?
68%, 13.5%, 95%
We produce 10,000 sticks of gum each month. The average stick of gum retains its flavor for 2 hours
with a standard deviation of 10 minutes. About how many sticks of gum will remain flavorful for: under
2 hours? under 100 minutes? over 130 minutes? between 110 and 130 minutes?
5000, 250, 1600, 6800