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Unformatted text preview: Chapter 2  Summarizing Data What happens if we change some of the values in our list? How will the change affect
the mean? The median? The standard deviation? Example 14: Changing Units
Find the mean, median, and standard deviation for each of the following lists. [/27
M=7'
er: [HM
b) 15,19,18,16,17 (+10)
M1!
M=l7
6": [J—HL“
c)10,18,16,12,14 (x Z)
M: l‘l
M=lLl _ 6‘: 2. 87.3
d) 15,27,24,18,21 (x ”53> M: —Zl M = ’2] CT: H.243
e) 13,21,19,15,17» (x1 3+3)
' #2 l7 M: l7 0’: 2.828 What rules can we deduce from this example? . If adding (subtracting) the same number to each value in a data set:
M J(Cd/3?: add/Sub‘éf‘acf same number CS‘/‘S 1 (lo Chan ¢
. If multiplying each value by positive constant: ,
61/] z _ mq/il‘P/y by 54m: Poshfl'i/c Cori sigma!
o If multiplying each value by a negative constant: \
M‘ M/X‘ : MM prlx/ b\/ 54m: (127151 Ve consian‘é
c—/.5 : multiply by 4105. Value 01: 90451994 f
. If adding and multiplying, a shift and scale change: ’ A
MVM/Y : all changes cagapflyp ..
5/3 : on\\/ .qPﬁgc'leof by mH/l—Ll‘p ’I‘Ccﬁgl‘o/I El" 4b behalf: \/a /qe 5 , 30 @[email protected]?$13337.35303311333!!!iiiiiiiﬂiiial Chapter 2 — Summarizing Data I 2.4 The Normal Approximation for Data What if we want to look at intervals besides the one given in the Empirical Rule? We
can use standardized scores or zscores. Z—scores tell how many SDs a data value
is above or below the sample mean data value. raw score — mean zscore = ————_——_—
standard devnatlon Example 15: ACT vs SAT Abigail and Emily wanted to compare their mathematics standardized test scores, but
Abigail only took the ACT, she got a 25 on the math section, and Emily only took the
SAT, she got a 620 on the math section. Nationwide the math ACT scores followed a
normal curve with mean 21 and standard deviation 6 while the math SAT scores followed a normal curve with mean 500 and standard deviation .100. Using this
information, how do Abigail and Emily’s scores compare? ‘ ‘ ZB'VZ\ __
Abiﬁou I: Z '5— T = :2; = O .(o . ‘ (922) Bob _ (10 ,_ ’
Eml\\/: Z: '00 _ ,Iéo . ['2‘ EMMY GLOQ ‘better 1' Once we calculate the zscores, we can ﬁnd the area over z—scoreintervals using the
standard normal curve. Example 16: Areas Use the standard normal curve, Table E, to find the following areas.
a) The area less than 2. AM ls“ <5“ 200 => 0.61772.
b) TWEN‘j‘éNnHandz. Alba (£10501: 2.00  A(‘€4\ [e95 c‘F’Z.O<) cme area gfeatoer than 1.5. O' 0‘77"}  0‘071'8 :
l'Areq “laP4; 5? l ‘3 s l" al.4352—o ocp
. — . 4,
Area leﬁbgF ~i.=‘s : n
d) The area between 1.25 and 1.75.
Afeq lep‘l: OF [.713 , Area. Kept o'F‘ 1.25 31 Chapter 2 — Summarizing Data 1 
Normal Approximation for Data .1
In summary, we can estimate the proportion of the data that are in an interval of {1
interest in three steps: 1 Step 1. Draw a number line and shade the interval of interest. Step 2. Convert the interval endpoints into z—scores using the formula
' raw score — mean zscore 2 ——~—_———.
standard deVIatIon Step 3. Find the area under the standard normal curve over the zscore interval. Example 17: ACTvs SAT, revisited
Using the information given in Example 15, answer the following questions.
a) What percent of people did worse than Abigail on the math ACT? 2; ZEJO’Zl : (92: Area laneé o‘F O.(o7 : (9.7ng _ b) What percent of people did better than Emily on the math SAT? __ (920—500 __
'2' too ’ Ll Area kept mt _\.’L“ ownM
' f;, c) What was Emily’s percentile on the math SAT? Area legE o’F L1 1 Mat—lei
 88.5441 (if70’”; d) What score on the SAT corresponds to the 85 percentile? Afaot left o‘F 7: — 0.9500
”2"; L036»? X7520 :1.03(o7 :1) ix: (003.(o7)x éOL.’ e) What score on the ACT corresponds to the 20th percentile?
Area leQt o“F 52" o. 2000
14: ~ 0. 8H2. , 32 @ Chapter 2 — Summarizing Data Example 18: Heights of Men
The heights of young men are approximately normally distributed with mean 70 inches and standard deviation 3.5 inches. a) What percent of men are shorter than 75 inches? ,_ “7%”70
E» %.5_ ._ Lazar; Neat (arty? Wise :2: AW {mint 5%: we 2.— omega W b) What percent of men are taller than 60 inches? 3' ””5535, ‘1' Z.“8‘37€ c) What percent of men are between 70 and 80 inches tall?
‘10 ~ ‘70 = 70 w;
.5517“ 3.15 :0 2%: $2.55 :2,@%.t Afea Ea'iwecﬂ %\ amﬁi T27;
* AFCa left a? 21 metres ﬂailg 65; g ‘ :21'6163761 ~0.5» :EO.“—lqw?ﬁ\ 7 “lei/76367533} d) What percent of men are between 72 and 82 inches tall? , 72—70 ﬁ ” 52~7©
a; r 6.5 " p.57l‘i Z1’ 3%? :1? ”3.5{28$ ‘Afaat heft/06894 %\ an 2'2.
‘ :: Afecl [291019 65: Z73,” A(‘eq gaff {3: El 1‘1 acacia??? 0.’71‘b‘7‘—‘ azgq :{ZQHWOE e) What height corresponds to the 70th percentile?
Area (E‘p'é of Z 3 07030 av, oezH /
X“70 20,524 >2> X1 71.93%”
(«3.5 ' _ f) What height corresponds to the 50th percentile?
Afcta [94715 of 2.; 3"" C3 . 5&66 .5 Chapter 2 — Summarizing Data Boxplot: The FiveNumber Summary us quartiles. The quartiles Q1, Q2, and Q3 divide the data into four equal parts. 1. Sort the data from smallest to largest and compute the median. This is the second
quartile, Q2. 2. The ﬁrst quartile, Q, is the median of all the data to the left of Q2.
3. The third quartile, Q3, is the median of all the data to the right of Q2. Example 19: Exam scores
Find Q1, Q2, and Q3 for the following set of Exam scores. To compare data, we can draw side by side boxplots using ﬁve summary values.
1. 2.
3.
4.
5.
These five values are referred to as the ﬁvenumber summary and the corresponding boxplOt is called a fivenumber boxplot. If outliers are present,
denote the outliers using asterisks, (*), and denote potential outliers with circles, (°). In tead of using the mean and the standard deviation to summarize the data, we can if at!
l 84 88 68 68 95
81 79 81 79 82 ..
79 81 79 _ 9o 81 1' j e <5 68 7‘1 7 ‘1 mm CUTVODTMED’IAIO! the minimum value,
the first quartile, the median, the third quartile, and the maximum value. 34 ...
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 Spring '09
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