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Frequency Distribution Table

After the collecting stage, the data must be organized. A frequency distribution table can be used. It is used to rank the disorganized data from the highest to the lowest. A frequency distribution table is an organized tabulation of the number of individual scores located in each category on the scale of measurement. It contains at least two columns - one for the score categories (X) and another for the frequencies (f).

A frequency distribution can be represented in a number of ways. 
Group frequency table; Regular frequency table
Group frequency graph - Polygon; Pie Chart; The bar graph; Histogram

The bar graph is used when the data or measurements are from a nominal or an ordinal scale.
Histogram - Interval or Ratio data
Polygon - Interval or Ratio data

Group frequency distribution is used when the original scale of measurement consists of more categories than can be listed in a regular table, 20 or more categories is generally considered too much. The Simplest and the most complete way to present a set of N= 7 scores is to list each individual score. This will call for the use of a polygon.

Important Notes:
A grouped frequency distribution table is needed when the range of scores is large, causing a regular frequency table to have too many entries in the score categories (X column). The guidelines for a group frequency table includes:
Striving for approximately 10 rows in the table;
Interval width of 2,5,10,20,50,100 should be used - depending upon the number of rows chosen;
The first (lowest) value of each interval should be a multiple of the interval width;
List all intervals without skipping any. The top interval should contain the highest observed X value and the bottom interval should contain the lowest observed X value.
The real limits for a score are the boundaries located halfway between the score and the next higher (or lower) score.

Example: Organize the data below into a frequency distribution table and show percentage of each category.
8, 9, 8, 7, 10, 9, 6, 4, 9, 8,
7, 8, 10, 9, 8, 6, 9, 7, 8, 8

x f ρ
10 2 0.1
9 5 0.25
8 7 0.35
7 3 0.15
6 2 0.10
5 0 0
4 1 0.05


                      

If the data covers a wide range of values it is reasonable to group them in a frequency table.

Draw a group distribution table from the data below. 4,4,6,6,7,8,9,9,10,11,11,11,12,12,12,13


 

x limits f
12 - 13 11.5 - 13.5 4
10 - 11 9.5 - 11.5 5
8 - 9 7.5 - 9.5 3
6 - 7 5.5 - 7.5 3
4 - 5 3.5 - 5.5 2

GRAPH

                                                        CENTRAL TENDENCY

Researches prefer to  describe distributions by talking about the "shape", central tendency", and "variability".

  • Central tendency measure where the center of the distribution is located.
  • Variability tells whether the scores are spread or clustered together.

CENTRAL TENDENCY:

Finding a single score that is the most representative of a group of data is the goal here. Three commonly used technique for measuring central tendency are the mean, the median, and the mode.

  • The mean is the same as the average.
  • The median is the item that occupies the very middle position when the data (score) is ranked from the smallest to the largest.
  • The mode is the number with the highest frequency ( occurs often).

MEAN:
There are several ways to find the mean depending on the type of data.



Examples:

Given the population 3,7,4,6. Find the mean, ยต:

x = 3+7+4+6 = 20
N = 4

Given the sample, 2,4,6,8,5,11. Find the mean .

COMPUTING MEAN FROM FREQUENCY DISTRIBUTION TABLE
When a frequency table is given, the mean can be found from the formula:

MEDIAN:
It is used instead of the mean when

  1. there is a skewed distribution.
  2. it is an open-ended distribution.
  3. there is an undetermined score.
  4. there is an ordinal scale.
  5. the median is not affected by extreme numbers.

Determining the median:

  1. when the sample size is odd - arrange the data in ascending or descending order and take the middle score.
  2. When the sample size is even - arrange the data in ascending or descending order and take the two middle numbers, add them up and divide the sum by 2.

MODE:
The score or group with the highest frequency.

  1. It is preferred when the scores are measured on a nominal scale.
  2. It can be used on all scales.

Central Tendency and the shape of the distribution:
All 3 measures of central tendency relate in a way. All 3 can have the same value under the same circumstances. The shape of the distribution determines the relationship.

NEW TERMS AND CONCEPTS

Frequency distribution A tabulation of the number of individuals in each
category on the scale of measurement.
Grouped frequency distribution A frequency distribution where scores are
grouped into intervals rather than listed individual values.
Class interval A group of scores in a grouped frequency distribution.
Upper real limit The boundary that separates an interval from
the next highest interval.
Lower real limit The boundary that separates
an interval from the next lowest interval.
Histogram A graph showing a bar above each score
or interval so that the height of the bar corresponds to
the frequency and width extends the real limits.
Bar graph A graph showing a bar above each score or inte rval
so that the height of the bar corresponds to
the frequency. A space is left between adjacent bars.
Polygon A graph consisting of a line that connects a series
of dots. A dot is placed above each score or interval
so that the height of the dot corresponds to the frequency.
Symmetrical distribution A distribution where the left-hand side is a mirror
image of the right-hand side.
Positively skewed distribution A distribution where the scores pile up on the
left side and taper off to the right.
Negatively skewed distribution A distribution where the scores pile up on
the right side and taper off to the left.
Tail(s) of a distribution A section on either side of a distribution
where the frequency tapers down toward zero
as the X values become more extreme.




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