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Frequency Distributions
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Frequency Distributions
After collecting data, the first task for aresearcher is to organize and simplify thedata so that it is possible to get a general
overvie of the results! "his is the goal of descriptive statistical
techniques!
#ne method for simplifying and organizingdata is to construct a frequencydistribution!
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Frequency Distributions %cont!&
A frequency distributionis an organized
tabulation shoing e'actly ho many
individuals are located in each category on
the scale of measurement! A frequency
distribution presents an organized picture
of the entire set of scores, and it shos
here each individual is located relative toothers in the distribution!
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Frequency Distribution "ables
A frequency distribution tableconsists of atleast to columns ) one listing categories on thescale of measurement %*& and another forfrequency %f&!
+n the * column, values are listed from thehighest to loest, ithout skipping any!
For the frequency column, tallies are determinedfor each value %ho often each * value occurs in
the data set&! "hese tallies are the frequenciesfor each * value!
"he sum of the frequencies should equal !
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Frequency Distribution "ables %cont!&
A third column can be used for theproportion %p& for each category. p / f0!"he sum of the p column should equal
1!! A fourth column can display the
percentage of the distribution
corresponding to each * value! "hepercentage is found by multiplying p by1! "he sum of the percentage column is1!
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4egular Frequency Distribution
5hen a frequency distribution table lists
all of the individual categories %* values& it
is called a regular frequency
distribution!
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7rouped Frequency Distribution
8ometimes, hoever, a set of scores
covers a ide range of values! +n these
situations, a list of all the * values ould
be quite long ) too long to be a 9simple:
presentation of the data!
"o remedy this situation, a grouped
frequency distributiontable is used!
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7rouped Frequency Distribution %cont!&
+n a grouped table, the * column lists groups of
scores, called class intervals, rather than
individual values!
"hese intervals all have the same idth, usuallya simple number such as 2, , 1, and so on!
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Frequency Distribution 7raphs
+n a frequency distribution graph, the
score categories %* values& are listed on
the * a'is and the frequencies are listed
on the > a'is!
5hen the score categories consist of
numerical scores from an interval or ratio
scale, the graph should be either ahistogram or a polygon!
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?istograms
+n a histogram, a bar is centered above
each score %or class interval& so that the
height of the bar corresponds to the
frequency and the idth e'tends to the
real limits, so that ad@acent bars touch!
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olygons
+n a polygon, a dot is centered above
each score so that the height of the dot
corresponds to the frequency! "he dots
are then connected by straight lines! An
additional line is dran at each end to
bring the graph back to a zero frequency!
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Bar graphs
5hen the score categories %* values& are
measurements from a nominal or an
ordinal scale, the graph should be a bar
graph!
A bar graphis @ust like a histogram e'cept
that gaps or spaces are left beteen
ad@acent bars!
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4elative frequency
Cany populations are so large that it is
impossible to kno the e'act number of
individuals %frequency& for any specific
category!
+n these situations, population distributions
can be shon using relative frequency
instead of the absolute number ofindividuals for each category!
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8mooth curve
+f the scores in the population are
measured on an interval or ratio scale, it is
customary to present the distribution as a
smooth curverather than a @aggedhistogram or polygon!
"he smooth curve emphasizes the fact
that the distribution is not shoing thee'act frequency for each category!
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Frequency distribution graphs
Frequency distribution graphs are usefulbecause they sho the entire set ofscores!
At a glance, you can determine thehighest score, the loest score, andhere the scores are centered!
"he graph also shos hether the scoresare clustered together or scattered over aide range!
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8hape
A graph shos the shapeof the distribution!
A distribution is symmetricalif the left side ofthe graph is %roughly& a mirror image of the right
side! #ne e'ample of a symmetrical distribution is the
bell)shaped normal distribution!
#n the other hand, distributions are skewed
hen scores pile up on one side of thedistribution, leaving a tail of a fe e'tremevalues on the other side!
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ositively and egatively
8keed Distributions
+n a positively skeweddistribution, the
scores tend to pile up on the left side of
the distribution ith the tail tapering off tothe right!
+n a negatively skeweddistribution, the
scores tend to pile up on the right side andthe tail points to the left!
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ercentiles, ercentile 4anks,
and +nterpolation
"he relative location of individual scores
ithin a distribution can be described by
percentiles and percentile ranks!
"he percentile rankfor a particular *
value is the percentage of individuals ith
scores equal to or less than that * value!
5hen an * value is described by its rank,
it is called a percentile!
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ercentiles, ercentile 4anks,
and +nterpolation %cont!& "o find percentiles and percentile ranks, to ne
columns are placed in the frequency distribution table.#ne is for cumulative frequency %cf& and the other is forcumulative percentage %c&!
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+nterpolation
5hen scores or percentages do not correspond
to upper real limits or cumulative percentages,
you must use interpolation to determine the
corresponding ranks and percentiles! Interpolationis a mathematical process based
on the assumption that the scores and the
percentages change in a regular, linear fashion
as you move through an interval from one end tothe other!
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8tem)and)Eeaf Displays
A stemandleaf displayprovides a veryefficient method for obtaining and displaying afrequency distribution!
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