CONSTRUCTION OF A FREQUENCY DISTRIBUTION A grouping of data into categories showing the number of observations in each mutually exclusive category.

DISCRETE FREQUENCY DISTRIBUTION The details regarding family number of children is given below, construct frequency distribution for the following data 2,3,4,1,1,0,2,3,4,0,1,3,2,4,0,1,2,2,2,3,2,3,1,1,1,0,1,2,3

5 Steps To Organize Raw Data Into A Frequency Distribution Statistics Is Fun! 5 Steps To Organize Raw Data Into A Frequency Distribution Step 1: Decide on Number of Classes Step 2: Determine The Class Interval Step 3: Set The Individual Class Limits Step 4: Tally The Data Into Classes Step 5: Count The Tallies in Each Class & Present the Frequency Distribution Statistics Are Fun!

Continuous Frequency Distribution Ms. Kathryn Ball of AutoUSA wants to develop tables, charts, and graphs to show the typical selling price on various dealer lots. The table on the right reports only the price of the 80 vehicles sold last month at Whitner Autoplex.

EXAMPLE – Constructing Frequency Distributions: Quantitative Data

Constructing a Frequency Table - Example Step 1: Decide on the number of classes. A useful recipe to determine the number of classes (k) is the “2 to the k rule.” such that 2k > n. There were 80 vehicles sold. So n = 80. If we try k = 6, which means we would use 6 classes, then 26 = 64, somewhat less than 80. Hence, 6 is not enough classes. If we let k = 7, then 27 128, which is greater than 80. So the recommended number of classes is 7. Step 2: Determine the class interval or width. The formula is: i  (H-L)/k where i is the class interval, H is the highest observed value, L is the lowest observed value, and k is the number of classes. ($35,925 - $15,546)/7 = $2,911 Round up to some convenient number, such as a multiple of 10 or 100. Use a class width of $3,000

Step 3: Set The Individual Class Limits Statistics Is Fun! Step 3: Set The Individual Class Limits Classes must be mutually exclusive Avoid overlapping or unclear class limits: Include lower limit Exclude upper limit Example of class limits: $12,000 up to $15,000 and $15,000 up to $18,000 $12,000 & $14,999 belong in the first class $15,000 belongs in the second class Avoid open ended classes (problems with graphing) The lower limit of the first class should be a multiple of the class interval (not always possible) Convenient multiples of ten are useful You must compare the actual range to the range implied by the number of classes & class interval General guidelines that are not always possible to follow. Thus, making Frequency Distributions is often refer to as an “art”. Statistics Are Fun!

Constructing a Frequency Table - Example

Constructing a Frequency Table Step 4: Tally the vehicle selling prices into the classes. Step 5: Count the number of items in each class.

Relative Frequency Distribution To convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of observations.

Class Intervals and Midpoints Class midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits. Class frequency: The number of observations in each class. Class interval: The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class.

Inclusive and Exclusive Inclusive Classes: If upper limit of the classes are included in the class itself then that classes are called Inclusive classes. Exclusive Classes: If upper limit of the classes are not included in the class itself then that classes are called Exclusive classes.

Class Boundaries and Class Limits Class Boundaries: If upper limit of one class is equal to lower limit of next class then that is called class Boundaries. Class Limits: If upper limit of one class is not equal to lower limit of next class then that is called class Limits.

Grouped Frequency Distributions Continuous Ratio/Interval Data Divide the overall range of the Observed values into classes; Count the number of observations that fall into each class. Example: Kenkel, pg.46, Monthly Salaries of employees at Lang’s Trucking Company

Histogram (For Quantitative data) A frequency distribution based on quantitative data A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other.

Frequency Polygon A frequency polygon also shows the shape of a distribution and is similar to a histogram. It consists of line segments connecting the points formed by the intersections of the class midpoints and the class frequencies.

Cumulative Frequency Distribution

Cumulative Frequency Distribution

Stem-and-Leaf Plots Stem-and-Leaf Plots: A convenient method to display every piece of data by showing the digits of each number.

Stem and Leaf Plot – a visual way to display a large set of data Stem – largest place value(s) of a number Leaf – smallest place value of a number

Statistics: Stem-and-Leaf Plots In a stem-and-leaf plot, the greatest common place value of the data is used to form stems. The numbers in the next greatest place value position are then used to form the leaves.

Step 1: Find the least and the greatest number in the set of data. 2 18 23 14 13 4 5 9 11 13 21 3 19 8 16 9 10 3 3 10 12 15 4 16 19

Step 2: Draw a vertical line between the digit in the smallest place value of the number and the other digits of the number. This gives the high and low for the range of the stems. least number 2 greatest number 2 3

Step 3: Make a chart with the titles STEM and LEAF. STEM LEAF

Step 4: Write the digits that form the stem in the STEM column. STEM LEAF 0 1 2

Step 5: Write the digits that form the leaf for each number in the LEAF column across from the STEM of the number. STEM LEAF 2 4 5 9 3 8 9 3 3 4 2 18 23 14 13 4 5 9 11 13 21 3 19 8 16 9 10 3 3 10 12 15 4 16 19 1 8 4 3 1 3 9 6 2 5 6 9 2 3 1

Age of United states Presidents at their First Inauguration: 57 61 57 57 58 57 61 54 68 51 49 64 50 48 65 52 56 46 54 49 50 47 55 54 42 51 56 55 51 54 51 60 62 43 55 56 61 52 69 64 (Through the 40th presidency) Stem Leaf 4 5 6 9 8 6 9 7 2 3 7 7 7 8 7 4 1 2 6 4 5 4 1 6 5 1 4 1 5 6 2 1 1 8 4 5 2 1 9 4

Statistics: Stem-and-Leaf Plots Age of United states Presidents at their First Inauguration: 57 61 57 57 58 57 61 54 68 51 49 64 50 48 65 52 56 46 54 49 50 47 55 54 42 51 56 55 51 54 51 60 62 43 55 56 61 52 69 64 (Through the 40th presidency) Rearrange the leaf in numerical order from least to greatest Stem Leaf 4 5 6 2 3 6 7 8 9 9 0 0 1 1 1 1 2 2 4 4 4 4 5 5 5 6 6 6 7 7 7 7 8 0 1 1 1 2 4 4 5 8 9

Statistics: Stem-and-Leaf Plots It is easy to interpret or analyze information from the Stem-and-Leaf. How many presidents were 51 years old at their inauguration? What age is the youngest president to be inaugurated? What is the age of the oldest president to be inaugurated? How many presidents were 40-49 years old at their inauguration? 4 42 69 7 Stem Leaf: Age of United States Presidents at their First Inauguration (through the 40th Presidency) 4 5 6 2 3 6 7 8 9 9 0 0 1 1 1 1 2 2 4 4 4 4 5 5 5 6 6 6 7 7 7 7 8 0 1 1 1 2 4 4 5 8 9