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TWO TYPES OF STATISTICS
• Descriptivestatistics summarizeor
describe theimportantcharacteristicsof
data.
• Inferentialstatistics usesampledatato
makeinferences (orgeneralizations)about
apopulation.
Sections 2-1 and 2-2
Review and Preview
and
Frequency Distributions
IMPORTANT CHARACTERISTICS
OF DATA
FREQUENCY DISTRIBUTION
1. Center: Arepresentativeoraveragevaluethat
indicateswherethemiddleofthedatasetis.
2. Variation: Ameasureoftheamountthatthe
datavaluesvaryamongthemselves.
3. Distribution: Thenatureorshapeofthe
distributionofthedata(suchasbell‐shaped,
uniform,orskewed).
4. Outliers: Samplevaluesthatlieveryfaraway
fromthevastmajorityoftheothersamplevalues.
5. Time: Changingcharacteristicsofdataovertime.
Afrequencydistribution (orfrequency
table)showshowadatasetispartitioned
amongallofseveralcategories(orclasses)by
listingallofthecategoriesalongwiththe
number(frequency)ofdatavaluesineachof
them.
QWERTY KEYBOARD WORD
RATINGS
AnarticleinDiscover magazine(“Typecasting”by
ScottKim,Discover)suggeststhatyoucan
measuretheeaseoftypingbyusingthispoint
ratingsystem:
Counteachletteronthehomerowas0,each
letteronthetoprowas1,andeachletteron
thebottomrowas2.
Togetthewordrating,
EXAMPLE:
wesumthenumbers.
ST ATI STI CS
Thewordratingfor
0101101120
“statistics”is7.
QWERTY KEYBOARD WORD
RATINGS
2
2
5
1
2
6
3
3
4
2
4
0
5
7
7
5
6
6
8
10
7
2
2
10
5
8
2
5
4
2
6
2
6
1
7
2
7
2
3
8
1
5
2
5
2
14
2
2
6
3
1
7
Thistableshowsthewordratingsforeachofthe52
wordsinthePreambletotheConstitution.
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FREQUENCY DISTRIBUTION OF
QWERTY WORD RATINGS
Rating
Lowerclasslimits arethesmallestnumbers
thatcanbelongtodifferentclasses.
Frequency
0-2
20
3-5
14
6-8
15
9 - 11
2
12 - 14
1
Rating
Lower Class
Limits
UPPER CLASS LIMITS
Upper Class
Limits
Frequency
0-2
20
3-5
14
6-8
15
9 - 11
2
12 - 14
1
CLASS BOUNDARIES
Upperclasslimits arethelargestnumbers
thatcanbelongtodifferentclasses.
Rating
LOWER CLASS LIMITS
Frequency
0-2
20
3-5
14
6-8
15
9 - 11
2
12 - 14
1
Classboundaries arethenumbersusedto
separateclasses,butwithoutthegapscreated
byclasslimits.
Rating
Frequency
- 0.5
2.5
Class
Boundaries
5.5
0-2
20
3-5
14
6-8
15
8.5
11.5
9 - 11
2
12 - 14
1
14.5
FINDING CLASS BOUNDARIES
Tofindclassboundaries:
1. Findthesizeofthegapbetweentheupper
classlimitofoneclassandthelowerclass
limitofthenextclass.
2. Addhalfofthatamounttoeachupperclass
limittofindtheupperclassboundaries.
3. Subtracthalfthatamountfromeachlower
classlimittofindthelowerclass
boundaries.
CLASS MIDPOINTS
Classmidpoints arethevaluesinthemiddle
oftheclasses.
Rating
Class
Midpoints
Frequency
0- 1 2
20
3- 4 5
14
6- 7 8
15
9 - 10 11
2
12 - 13 14
1
2
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CLASS WIDTH
Classwidth isthedifferencebetweentwo
consecutivelowerlimits(ortwoconsecutive
lowerclassboundaries)inafrequency
Rating
Frequency
distribution.
Class Width
3
0-2
20
3
3-5
14
3
6-8
15
3 9 - 11
2
3 12 - 14
1
RELATIVE FREQUENCY
DISTRIBUTION
frequencyforaclass
sumofallfrequencies
percentagefrequency
frequencyforaclass
sumofallfrequencies
100%
CUMULATIVE FREQUENCY
DISTRIBUTION
Acumulativefrequency foraclassisthesum
ofthefrequenciesforthatclassandallprevious
classes.
Rating Frequency
Rating
Cumulative
Frequency
0-2
20
Less than 3
20
3-5
14
Less than 6
34
6-8
15
Less than 9
49
9 - 11
2
Less than 12
51
12 - 14
1
Less than 15
52
1. Selectthenumberofclasses,usuallybetween5and20.
2. Determinetheclasswidth
maximumdatavalue
minimumdatavalue
classwidth
numberofclasses
Roundthisresulttogetaconvenientnumber.(It’susuallybestto
roundup.)
3. Choosethevalueforthefirstlowerclasslimitbyusingeitherthe
minimumvalueoraconvenientvalueslightlybelowtheminimum.
4. Usingthefirstlowerclasslimitandtheclasswidth,listtheotherlower
classlimits.(Addtheclasswidthtothefirstlowerclasslimittogetthe
secondlowerclasslimit.Addthewidthtothesecondlowerlimitto
getthethird,andsoon.)
5. Listthelowerclasslimitsinaverticalcolumnandentertheupper
classlimits.
6. Takeeachindividualdatavalueandputatallymarkintheappropriate
class.Addthetallymarkstofindthetotalfrequencyforeachclass.
RELATIVE FREQUENCY
DISTRIBUTION
Arelativefrequencydistribution or
percentagefrequencydistribution includes
thesameclasslimitsasafrequencydistribution,
butrelativefrequenciesareusedinsteadofactual
frequencies.
relativefrequency
CONSTRUCTING A FREQUENCY
DISTRIBUTION
Cumulative
Frequencies
Rating Frequency
Relative
Rating Frequency
0-2
20
0-2
38.5%
20/52=38.5%
3-5
14
3-5
26.9%
14/52=26.9%
6-8
15
6-8
28.8%
9 - 11
2
9 - 11
3.8%
12 - 14
1
12 - 14
1.9%
etc.
Totalfrequency=52
CRITICALLY THINKING: USING
FREQUENCY DISTRIBUTIONS TO
UNDERSTAND DATA
Inlaterchapters,therewillbefrequentreference
todatawithanormaldistribution.Onekey
characteristicofanormaldistributionisthatit
hasa“bell”shape.
• Thefrequenciesstartlow,thenincreasetoone
ortwohighfrequencies,thendecreasetoalow
frequency.
• Thedistributionshouldbeapproximately
symmetric,withfrequenciesprecedingthe
maximumbeingroughlyamirrorimageof
thosethatfollowthemaximum.
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