Chapter 2: Frequency Distribution

A set of scores if presented in a frequency distribution histogram. I fthe histogram shows a series of bars that tend to decrease in height from left to right, then what is the shape of the distribution?

positively skewed

For the scaores shown in the following stem and leaf display, how many people had scores in the 70s? 9/374 8/945 7/7042 6/68 5/14

4

For the scores shown in the following stem and leaf display, what is the lowest score in the distribution? 9/374 8/945 7/7042 6/68 5/14

51

Step Two: Locate the position of the intermediate value in the interval. This position corresponds to a fraction of the whole interval.

fraction = distance from the top of the interval divided by the interval width

Interpolation

is 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 to the other.

frequency distribution

is an organized tabulation showing exactly how many individuals are located in each category on the scale of measurement.

One method for simplifying and organizing data

is to construct a frequency distribution.

After collecting data, the first task for a researcher

is to organize and simplify the data so that it is possible to get a general overview of the results. ***•This is the goal of descriptive statistical techniques.***

Variability

measures the degree to which the scores are spread over a wide range or are clustered together.

When the exact number of individuals is not known

population distributions can be shown using relative frequency instead of the absolute number of individuals for each category

Frequency Distribution Tables •A third column can be used for the proportion (p) for each category: p = f/N.

-Because proportions describe the frequency (f) in relation to the total number (N), they often are called relative frequencies. -The sum of the p column should equal 1.00.

•The interpolation process requires four steps.

-Find the width of the interval on both score scale and percentage scale. -Locate the position of the intermediate value in the interval. This position corresponds to a fraction of the whole interval. -Use the same fraction to determine the corresponding position on the other scale. -Use the distance from the top to determine the position on the other scale.

•A frequency distribution table consists of at least two columns- one listing categories on the scale of measurement (X) and another for frequency (f).

-In the X column, values are listed from the highest to lowest, without skipping any. -For the frequency column, tallies are determined for each value (how often each X value occurs in the data set). -The sum of the frequencies should equal N.

•When you can obtain an exact frequency for each score in a population, you can construct frequency distribution graphs that are exactly the same as the histograms, polygons, and bar graphs that are typically used for samples.

-Many populations are so large that it is impossible to know the exact number of individuals (frequency) for any specific category.

Frequency Distribution Tables A fourth column can display the percentage of the distribution corresponding to each X value

-The percentage is found by multiplying p by 100. -The sum of the percentage column is 100%.

central tendency

-measures where the center of the distribution is located.

A frequency distribution can be contructed as a

-table or as a graph, and presents the same two elements: •The set of categories that make up the original measurement scale. •A record of the frequency, or number of individuals in each category.

Identify the intervals

1. Identify your lowest score: 53 2. The bottom score of the first interval has to be a multiple of the width and 53 has to be in the first interval 3. If we multiple 5 by 10 we get 50 so the bottom score of the interval is 50 4. The first interval is 50-54 5. The last is 90-94 (your highest score is 94)

When the percentage is given but the score is not

1.Width of score scale is (top - bottom) = 9.5-4.5, and width of percentage scale is (top - bottom) 60-10 2.Fraction = distance from top percentage / percentage width =(60-40)/50=20/50=0.4 3.Distance = fraction / Width of score scale = 0.4 / 5 = 2 4.Position on score scale = Top score (9.5) - percentage distance (2) = 7.5 The 40th percentile corresponds to X = 7.5

A set of scores ranges from a high frequency of X=48 to a low of X=13. If these scors are placed in a group frequency distribution table with an interval width of 5 points, the bottom interval in the table would be -----?

10-15 width is 5 points and 13 falls within the interval

In a grouped frequency distribution one interval is listed as 20-24. Assuming that the scores are measuring a continuous variable, what is the width of this variable?

20=1,21=1,22=1,23=1,24=1 Total is 5

The seminar rooms in the library are identified by the letters A, b, c and so on. A professor records the number of classes held in each room during the first semester. If these values are presented in a frequency distribution graph, what kind of graph would be appropriate? a. Histogram b. Polygon c. A histogram or Polygon d. A bar Graph

A bar graph

positively skewed distribution

A distribution where the scores pile up on the left side and taper off to the right.

Example: An instructor has obtained the set of N=25 exam scores shown here. To help organize these scores, we will place them in a refquency distribution table. The scores are: 82,75,88,93,53,84,87,58,72,94,69,84,61,91,64,87,84,70,76,89,75,80,73,78,60

Calculate Range: (highest scores - lowest score)+1 (94-53)+1=42 Number of intervals needed to cover a Range of 42 Width 2 - 21 (too many) 5 - 9 (ok) 10 - 5 (too few)

Step Three: Use the same fraction to determine the corresponding position on the other scale.

Distance = fraction / width of percentage scale

Symmetrical Distribution Shape

Mound shaped

To find percentiles and percentile ranks, two new columns are placed in the frequency distribution table:

One is for cumulative frequency (cf) and the other is for cumulative percentage (c%). c%=(cf/N)(100%)

Step Four: Use the distance from the top to determine the position on the other scale.

Position on percentage scale= Top percentage - score distance = percentage X = number which is the ?th percentile

A group of quiz scores ranging from 4-9 are shown in a histogram. If the bars in the histogram gradually increase in height from left to right, what can you conclude about the set of quiz scores?

There are more high scores than there are low scores.

Step One: Find the width of the interval on both score scale and percentage scale.

Width of score scale is top minus bottom and width of percentage scale top minus bottom.

Example: The following set of N=20 scores was obtained from a 10-point statistics quiz. We will organize these scores by constructing a fequency distribution table. Scores: 8,9,8,7,10,9,6,4,9,8,7,8,10,9,8,6,9,7,8,8

X f 10 - 2 9 - 5 8 - 7 7 - 3 6 - 2 5 - 0 4 - 1

For the following frequency distribution, how many individuals had a score of X=2? x - f 5 - 1 4 - 2 3 - 4 2 - 3 1 - 2

X=2 is 3 individuals

The following is a distribution of quiz scores. If a score of X=2 or lower is failing, then how many infdividuals failed the quiz? x - f 5 - 1 4 - 2 3 - 4 2 - 3 1 - 2

X=2 or lower is 3+2=5

For this distribution, how many individuals had scores lower than X=20? X = f 24-25 = 2 22-23 = 4 20-21 = 6 18-19 = 3 16-17 = 1

You count all individuals with scores less than 20 3+1=4

Stem-and-Leaf Displays

are like histograms, but they also give the individual values. -Each score is divided into a stem consisting of the first digit or digits, and a leaf consisting of the final digit. -Then, go through the list of scores, one at a time, and write the leaf for each score beside its stem.

Frequency Distribution Graphs (interval and ratio data)

•In a frequency distribution graph, the score categories (X values) are listed on the X axis and the frequencies are listed on the Y axis. •When the score categories consist of numerical scores from an interval or ratio scale, the graph should be either a histogram or a polygon.

grouped frequency distribution

•Sometimes a set of scores covers a wide range of values (i.e. X = 40 to X =80) -A list of all the X values would be too long to be a "simple" presentation of the data. •In a grouped table, the X column lists groups of scores, called class intervals, rather than individual values.

Grouped Frequency Distribution •The grouped frequency distribution table should have about 10 class intervals. •The width of each interval should be a relatively simple number such as 2, 5, 10 or 20.

•The bottom score in each class interval should be a multiple of the width. •All intervals should be the same width.

percentile rank

•The percentile rank for a particular X value is the percentage of individuals with scores equal to or less than that X value. •When an X value is described by its rank, it is called a percentile.

Percentiles, Percentile Ranks, and Interpolation

•The relative location of individual scores within a distribution can be described by percentiles and percentile ranks.

If the scores in the population are measured on an interval or ratio scale, it is customary to present the distribution as a smooth curve

•The smooth curve emphasizes the fact that the distribution is not showing the exact frequency for each category.

•Researchers often simply describe a distribution by listing its characteristics.

•There are three characteristics that completely describe any distribution: shape, central tendency, and variability.

Example: •Each cumulative percentage identifies the percentile rank for the upper real limit of the corresponding score or class interval.

•X = 4.5 is the 95th percentile

Frequency Distribution Graphs (interval and ratio data) ***Histogram***

•a bar is centered above each score (or class interval). -The height of the bar corresponds to the frequency. -The width extends to the real limits, so that adjacent bars touch.

Frequency Distribution Graphs (interval and ratio data) ***Polygon***

•a dot is centered above each score. -The height of the dot corresponds to the frequency. -A continuous line is drawn from dot to dot to connect the series of dots. -The graph is completed by drawing a line down to the X-axis (zero frequency) at each end of the range of scores.

Frequency Distribution Graphs (nominal or ordinal data) ***Bar Graph***

•is just like a histogram except that gaps or spaces are left between adjacent bars. -For a nominal scale, the space emphasizes that the scale consists of separate, distinct categories. -For ordinal scales, separate bars are used because you cannot assume that the categories are all the same size. -Has space between the bars.

shape

•shows the shape of the distribution. •A distribution is symmetrical if the left side of the graph is (roughly) a mirror image of the right side. •Distributions are skewed if the scores tend to pile up toward one end of the scale and taper off gradually at the other end. -The section where the scores taper off toward one end of a distribution is called the tail of the distribution.

negatively skewed distribution

•the scores tend to pile up on the right side and the tail points to the left.