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# Freq Distributions Math

Feb 23, 2018

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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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1(

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

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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 stem-and-leaf displayprovides a veryefficient method for obtaining and displaying afrequency distribution!

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