Frequency Distributions

Introduction to Research Design and Statistics

Frequency Distributions

Data Analysis

Data analysis is mostly about organizing quantitative information (a bunch of numbers) so that it is more understandable or more easily interpreted.

Example Data #1

A Bunch of Numbers

41 53 54 47 57 51 42 45 54 52 51 51 71 57 50 43 51 60 62 57
35 75 45 57 45 46 66 57 49 49 57 64 63 57 50 58 75 68 44 40
41 62 57 43 48 63 39 70 63 59 61 38 61 49 73 44 42 39 55 52
45 61 39 41 50 40 60 47 59 49 46 58 71 58 46 43 54 56 46 54
57 54 71 48 40 64 51 39 40 61 66 49 65 52 46 61 72 71 40 69
64 56 49 54 53 66 67 40 46 69 40 41 57 58 57 37 55 62 64 40
50 46 53 52 45 56 45 54 56 41 54 72 56 47 49 60 54 55 33 49
43 50 52 48 58 43 41 43 46 44 43 61 40 49 56 61 50 51 42 67
53 50 51 72 48 40 53 39 63 51 45 39 42 62 44 65 63 54 45 60
49 48 57 55 66 64 55 42 56 53 41 42 53 42 60 52 38 57 58 65

Frequency Distribution

A tabular display of values and their associated frequencies.

A Frequency Distribution for Ungrouped Data


                      Cum.      Cum.
  score  Freq.   %    Freq.      %
-------------------------------------
    75    2     1.0    200     100.0  
    73    1     0.5    198      99.0  
    72    3     1.5    197      98.5  
    71    4     2.0    194      97.0  
    70    1     0.5    190      95.0  
    69    2     1.0    189      94.5  
    68    1     0.5    187      93.5  
    67    2     1.0    186      93.0  
    66    4     2.0    184      92.0  
    65    3     1.5    180      90.0  
    64    5     2.5    177      88.5  
    63    5     2.5    172      86.0  
    62    4     2.0    167      83.5  
    61    7     3.5    163      81.5  
    60    5     2.5    156      78.0  
    59    2     1.0    151      75.5  
    58    6     3.0    149      74.5  
    57   13     6.5    143      71.5  
    56    7     3.5    130      65.0  
    55    5     2.5    123      61.5  
    54   10     5.0    118      59.0  
    53    7     3.5    108      54.0  
    52    6     3.0    101      50.5  
    51    8     4.0     95      47.5  
    50    7     3.5     87      43.5  
    49   10     5.0     80      40.0  
    48    5     2.5     70      35.0  
    47    3     1.5     65      32.5  
    46    8     4.0     62      31.0  
    45    8     4.0     54      27.0  
    44    4     2.0     46      23.0  
    43    7     3.5     42      21.0  
    42    7     3.5     35      17.5  
    41    7     3.5     28      14.0  
    40   10     5.0     21      10.5  
    39    6     3.0     11      5.5  
    38    2     1.0      5      2.5  
    37    1     0.5      3      1.5  
    35    1     0.5      2      1.0  
    33    1     0.5      1      0.5

Frequency Histogram

Histogram with Normal Curve

Histogram with Boxplot

Kernal Density Plot

Frequency Polygon

Frequency Polygon Superimposed on a Histogram

Relative Frequency Histogram

Stem-and-leaf Plot


  3* | 3
  3. | 5788999999
  4* | 00000000001111111222222233333334444
  4. | 5555555566666666777888889999999999
  5* | 00000001111111122222233333334444444444
  5. | 555556666666777777777777788888899
  6* | 00000111111122223333344444
  6. | 555666677899
  7* | 011112223
  7. | 55

Box Plot

Cumulative Frequency

Cumulative Relative Frequency

Example Data #2

A Frequency Distribution for Grouped Data


                         Cum      Real       Mid-
Score    f   cf    %      %      Limits     Point
 71-75  10  200   5.0  100.0   70.5-75.5     73
 66-70  10  190   5.0   95.0   65.5-70.5     68
 61-65  24  180  12.0   90.0   60.5-65.5     63
 56-60  33  156  16.5   78.0   55.5-60.5     58
 51-55  36  123  18.0   61.5   50.5-55.5     53
 46-50  33   87  16.5   43.5   45.5-50.5     48
 41-45  33   54  16.5   27.0   40.5-45.5     43
 36-40  19   21   9.5   10.5   35.5-40.5     38
 31-35   2    2   1.0   1.0    30.5-35.5     33

Frequency Histogram

Cumulative Frequency Histogram

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Phil Ender, 13Oct99