Math 211 Lecture II

Chapter 2 Frequency Distributions

Data: A collection of values to be used for statistical analysis.

Raw Data: Collected data which does not need to be numerical. i.e. set of weights of certain set of students, days of the week, etc.

Array: Arrangement of raw numerical data in ascending or descending order.

Range: Maximum data – minimum data.

Class Interval: A class interval is a division of data for use in Histogram(a type of Bar graph). For instance, it is possible to partition scores on a 100 point test into class intervals of 1-25, 26-49, 50-74 and 75-100. The end numbers are called class limits; the smaller numbers are Lower Class Limits (LCL) and the larger numbers are the Upper Class Limits (UCL). The numbers 0.5-25.5, 25.5-49.5, 49.5-74.5, 74.5-100.5 are called class boundaries. For example 0.5 is a lower class boundary and 25.5 is an upper class boundary of the first class.

Class Interval Size (widthness=c): Upper class boundary – lower class boundary.

Class Frequency: Number of individuals belonging to each class.

Class Mark(CM): The midpoint of the class interval.

or .

Frequency Tables (Frequency Distributions):

The first step in drawing a frequency distribution is to construct a frequency table. A frequency table is a way of organizing the data by listing every possible score (including those not actually obtained in the sample) as a column of numbers and the frequency of occurrence of each score as another. Simply a frequency distribution is an arrangement of data by classes together with the corresponding class frequency.

Computing the frequency of a score is simply a matter of counting the number of times that score appears in the set of data. It is necessary to include scores with zero frequency in order to draw the frequency polygons correctly.

General Rules for forming Frequency Distributions:

1. step: Find the range

2. step: Divide the range into a convenient number of class intervals having the same size (possible number of class intervals are changing between 5 and 20).

3.step: find the class frequencies.

Histograms:

A histogram is drawn by plotting the scores (midpoints) on the X-axis and the frequencies on the Y-axis. A bar is drawn for each score value, the width of the bar corresponding to the real limits of the interval and the height corresponding to the frequency of the occurrence of the score value

Frequency Polygons:

A frequency polygon is drawn exactly like a histogram except that points are drawn rather than bars. The X-axis begins with the midpoint of the interval immediately lower than the lowest interval, and ends with the interval immediately higher than the highest interval.

Relative Frequency of a class: It is a percentage which is obtained by dividing the frequency of the class to the total frequency of all classes.

Relative Frequency Distribution: Arrangement of data by classes together with the corresponding relative frequencies.

Cumulative Frequency: The total frequency of all values less than the upper class boundary of a given class interval is called the cumulative frequency upto and including that class interval.

Plotting scores on the X-axis and the cumulative frequency on the Y-axis draws the Ogive (cumulative frequency polygon). The points are plotted at the intersection of the upper class boundary of the interval and the cumulative frequency.

Relative Cumulative Frequency: This is also called the percentage cumulative frequency which is obtained by dividing the cumulative frequency to the total frequency.

Drawing the X-axis as before and the relative cumulative frequency on the Y-axis draws the Percentage Ogive (relative cumulative frequency polygon).

Example 1: The frequency distribution of the ages of a sample of 400 diabetics obtained by a research physician are given.

Age (years) / No. of
diabetics / Class marks
0-9 / 21 / 4.5
10-19 / 29 / 14.5
20-29 / 39 / 24.5
30-39 / 31 / 34.5
40-49 / 88 / 44.5
50-59 / 72 / 54.5
60-69 / 50 / 64.5
70-79 / 40 / 74.5
80-89 / 30 / 84.5
400

Construct

(a)a Frequency Histogram and a Frequency Polygon.

(b)a Relative Frequency Distribution.

(c)a Cumulative Frequency Distribution and an Ogive.

(d)a Relative Cumulative Frequency Distribution and a Percentage Ogive.

(e)Estimate the percentage of diabetics whose age is under 42.

Example 2: The following data represents the age of the buildings (in years) of a given area in Lefkosa.

10 / 12 / 11 / 11 / 18
14 / 37 / 15 / 27 / 21
15 / 7 / 2 / 12 / 11
15 / 25 / 4 / 3 / 9
16 / 10 / 24 / 2 / 20
18 / 3 / 12 / 29 / 12

(a)Construct the frequency table from this data using 6 classes.

(b)Draw the relative cumulative frequency Histogram and the Percentage Ogive.

(c)Estimate the percentage of houses whose age is under 15 years.

Example 3: Thirty AA Batteries were tested to determine how long they would last. The results, to the nearest minute, were recorded as follows.

423 / 387 / 393 / 371 / 389
392 / 431 / 363 / 405 / 369
411 / 394 / 377 / 409 / 408
401 / 391 / 382 / 400 / 381
399 / 415 / 428 / 422 / 396
372 / 410 / 419 / 386 / 390

Construct (a) a Frequency Distribution.

(b) a Cumulative Frequency Distribution.

Example 4: In a 2-weekly study of the productivity of workers, the following data were obtained on the total number of acceptable pieces which 40 workers produced.

65 / 36 / 49 / 84 / 43 / 78 / 37 / 40
50 / 60 / 56 / 59 / 48 / 76 / 74 / 88
35 / 62 / 52 / 63 / 76 / 60 / 48 / 55
21 / 35 / 61 / 45 / 45 / 53 / 34 / 67
62 / 65 / 55 / 61 / 41 / 74 / 82 / 58

(a)Construct a frequency distribution using 6 classes.

(b)Draw a frequency histogram.

(c)Construct a relative frequency histogram.

(d)Draw the percentage ogive.

Example 5: The inner diameters of washers produced by a company can be measured to the nearest hundredth of millimeter. If the class marks of a frequency distribution of these diameters are given in millimeters by 8.15, 8.22, 8.29, 8.36, 8.43, and 8.50, find

(a)the class interval size

(b)the class boundaries

(c)the class limits.

Example 6: The following table shows a frequency distribution of the weekly wages of 65 employees at the P&R Company.

Wages / No. of
employees
$250.00-259.99 / 8
$260.00-269.99 / 10
$270.00-279.99 / 16
$280.00-289.99 / 14
$290.00-299.99 / 10
$300.00-309.99 / 5
$310.00-319.99 / 2
N=65

(a)Construct a frequency histogram

(b)Construct a cumulative-frequency distribution

(c)Construct an ogive

(d)Evaluate the number of employees earning

(i)less than $288.00 per week

(ii)at least $263.00 per week but less than $275.00 per week.

1