Learning Outcomes:

• Understand the definition, concepts, and uses of frequency distribution;

• Know how to understand and interpret a frequency distribution;

• Know how to create a frequency distribution;

• Learn when to use various methods or ways of creating a frequency distribution.

CHAPTER 2 

FREQUENCY DISTRIBUTIONS

WHAT IS A FREQUENCY DISTRIBUTION?  

• Frequency distribution is an organized tabulation of the number of individuals located in different categories at different levels of measurement. 

• This is used to group scores together which allows researchers to get a glance at the set of scores. 

• Used to organize nominal-level type of data.

2.1 Frequency Distributions and Frequency Distribution Tables•

• Frequency  distribution:  organized  tabulation  showing  the  number  of  individuals located in each category on the scale of measurement

X heading for the scores

f heading for the frequencies

Proportions and Percentages

• proportion = p = f/N

• percentage = p(100) = f/N(100)

CONSTRUCT A FREQUENCY DISTRIBUTION FOR THE DATA 

Step 1: Arrange the data (ascending/descending).

Step 2: Determine the classes (range, # or classes, interval, CB)  

•  Find the highest and lowest value 

• Find the Range o Range = HV-LV

The range is the difference between the highest value and the lowest value in a distribution.  

• How to determine the number of classes: “2 to the k rule”   

• Determine the class interval (width) 

Class Interval is the distance between the lower-class boundary and the upper-class boundary; denoted by the "i" 

𝑟𝑎𝑛𝑔𝑒 𝐻𝑉−𝐿𝑉

𝑖 = ------------------------------- or 𝑖 = ----------------

  𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠 K

Class boundaries are the upper and lower values of a class whose values have additional decimal places more than the class limits and end with the digit, 5

• To obtain class boundaries, subtract 0.5 from each lower-class limit and add 0.5 to each upper-class limit.  

Step 3: Tally the raw data into numerical frequencies.

Step 4: Determine the relative frequency. 

• Relative frequency (rf) is the value obtained when the frequencies in each class are divided by the total number of values.  

Step 5: Determine the percentage

• The percentage is obtained by multiplying the relative frequency by 100%.  

Step 6: Determine the cumulative frequency.

• Cumulative frequency (cf) is the sum of the frequencies accumulated up to the upper boundary of a class in a frequency distribution.  

Step 7: Determine the Midpoints. 

• The midpoint is the point halfway between the class limits of each class and is representative of the data within that class. 

• Can be found by getting the average of the upper limit and lower limit in each class.

2.2 Grouped Frequency Distribution Tables

• about 10 class intervals

• the width should be a relatively simple number

• bottom score in each class interval should be a multiple of the width

• all intervals should be the same width

2.3  Frequency Distribution Graphs 

Graphs for Interval or Ratio Data

• Histograms: First  List the scores equally spaced along the  X-axis,  then  draw  a  bar above each X value so that: 

a. The height of the bar corresponds to the frequency for that category

b. For continuous variables,  the bar's width extends to the category's real limits.  

c. For  Discrete Variables,  each bar extends half the distance to the adjacent category on each side

• Polygons: List the scores equally spaced along the X-axis, then:  

a. a dot is centered above each score so that the vertical position corresponds to the frequency for the category. 

b. a continuous line is drawn from dot to dot to connect the series of dots. 

c. draw a line down to the X-axis at each end of the range of scores.

Graphs for Nominal or Ordinal Data

• bar graphs: similar to a histogram but a space is left between the bars.

Graphs for Population Distributions

• relative frequencies: does not show the number just the relative difference between categories

• smooth curves: shows the relative changes that occur from one score to the next-population distribution usually shown by a normal curve  (mathematically guaranteed).

The Shape of a Frequency Distribution

• shape, variability (the degree to which the scores are spread or clustered), and central tendency (where the center of the distribution is located) describe any distribution.

• symmetrical distribution:  it is possible to draw a vertical line through the middle so that one side of the distribution is a mirror image of the other.

• skewed distribution: the scores tend to pile up toward one end of the scale and taper off gradually at the other end. 

- where scores taper off is called the tail of the distribution

- tail on the right-hand side: positively skewed

- tail on the left-hand side: negatively skewed