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Freq Distributions Math

Feb 23, 2018

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Gay Delgado
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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

    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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    1;

    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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