The Confusion Between A Direct Relationship and A Linear Relationship Ted Mitchell

There are Two Definitions A Direct Relationship between two variables, X and Y, is one in which the ratio of the two variables has a constant value, k X/Y = k All Direct Relationships are on a line between the origin and the observation of a single performance, (X, Y) A Linear Relationship between two variables, X and Y, is one in which the ratio of the differences between two points of Observation, ∆X/∆Y, has a constant value, m ∆X/∆Y = m All Linear Relationships require two performances for their description

People Confuse Direct and Linear Relationships Several sources for this confusion: 1) All Direct Relationships are Linear Relationships. However, NOT all Linear Relationships are Direct Relationships

Linear Relationships and Direct Relationships Set of all Linear Relationships Direct Relationships that are Linear

A Direct Relationship between Number of Salespeople, S, and Quantity sold, Q 0, 0 Number of Salespeople or Servers, S o Amount of Product Sold, Q Slope of the line = Q/S and the ratio has a constant value Q/S = k All Direct Relationships Pass though the origin, (0, 0)

Because all Direct Relationships pass through the origin, you only need one observed performance to describe it 0, 0 Number of Salespeople or Servers, S o Amount of Product Sold, Q A single point of observation (S. Q) A single point of observation (S. Q) All Direct Relationships Pass though the origin, (0, 0)

A Linear relationship must have at least two points of observed performance 0, 0 Number of Salespeople, S Amount of Product Sold, Q a All Direct Relationships Pass though the origin, (0, 0) Observed point, (S, Q) Observed point, (S 1, Q 1 ) Observed point, (0, a)

A Linear relationship is defined by the ratio of the differences having a constant value, ∆Q/∆S = m 0, 0 Number of Salespeople, S Amount of Product Sold, Q a All Direct Relationships Pass though the origin, (0, 0) Observed point, (S, Q) Observed point, (S 1, Q 1 ) ∆S ∆Q

The Definition of a Linear relationship is that the ratio of the differences, ∆Q/QS, between the two points of observations, (S1, Q1) and (S, Q), has a constant value, m 0, 0 Number of Salespeople, S Amount of Product Sold, Q The slope of the relationship ∆Q/∆S = m a All Direct Relationships Pass though the origin, (0, 0)

The line of a Linear relationship does not have to pass through the origin 0, 0 Number of Salespeople, S Amount of Product Sold, Q Q = a + (∆Q/∆S)S Q = a + mS a All Direct Relationships Pass though the origin, (0, 0)

The relationship between the amount sold, Q, and the number of servers, S, can be Linear and NOT be a Direct relationship 0, 0 Number of Salespeople, S Amount of Product Sold, Q Q = a + (∆Q/∆S)S Q = a + mS a All Direct Relationships Pass though the origin, (0, 0)

The Direct Relationship between Number of Salespeople, S, and Quantity sold, Q, is a Linear Relationship because there are Two Points of performance 0, 0 Number of Salespeople or Servers, S o Amount of Product Sold, Q Observed point, (S, Q) Observed point, (0, 0)

A Direct Relationship between Number of Salespeople, S, and Quantity sold, Q, is a Linear Relationship because there are Two Points of Performance 0, 0 Number of Salespeople or Servers, S o Amount of Product Sold, Q Observed point, (S, Q) Observed point, (0, 0) ∆S ∆Q

We can forecast a New Output, Q 1, from the single performance of a Direct relationship given a newly proposed amount of input, S 1 because direct relationships are also linear relationships Q 1 = k x S 1 k = Q/S = ∆Q/∆S = (Q–0)/(S–0) This a special characteristic All Direct Relationships are linear relationships

We can forecast a Linear Relationships can only claim that the ratio of the two differences is constant ∆Q/∆S = m

The practical importance of the two definitions is That Marketing managers can use direct relationship between the two observed differences, ∆Q and ∆S, of a linear relationship for making a forecast about the size of future changes Forecast the size of the change in demand, ∆Q 1, given a proposed change in the number of servers, ∆S 1 ∆Q 1 = (∆Q/∆S) x ∆S 1 ∆Q 1 = m x ∆S 1

#1 Example There is a direct relationship between the number of hours a café is open, H, and the number of cups sold, Q. Your shop is normally open 100 hours a week and sells 2,000 cups of coffee in a week. The rate at which the café converts hours into demand is considered to have a constant value Q/H = 2,000 cups/100 hours = 20 cups per hour When the café is kept open for 105 hours the coffee sales are 2,200 cups per hour. 1) Because it is a direct relationship the conversion rate is constant Q/H = 2,100 cups/105 hours = 20 cups per hour 2) Because it is a linear relationship the ratio of difference in the cups sold, ∆Q =100 cups, and ∆H = 5 hours has a constant value of ∆Q/∆H = 100 cups /5 hours = 20 cups per hour 3) Because it is both a direct relationship and a linear relationship the ratio of the variables, Q/H, and the ratio of the change in the variables ∆Q/∆H are equal to each other. Q/H = ∆Q/∆H = 20 cups per hour The assumption of a direct relationship is reasonable because a closed café (0 hours) sells a zero number of cups.

A Direct Relationship is one where the ratio of the two variables has a constant value, Q/H = k 0, 0 Number of Hours, H Amount of Product Sold, Q ∆H = ∆Q = 2,100 – 2, hours 105 hours

#1 Example of Two Performances in a Direct Relationship Defining the Linear Relationship Direct Relationship Performance 1 Direct Relationship Performance 2 Linear relationship ∆ performance = 2-1 Hours, HH 1 = 100H 2 = 105∆H = 5 Q/H = Cups per hour Q/H = 2,000/100 = 20 cph Q/H = 2,000/105 =20 cph ∆Q/∆H =200/5 ∆Q/∆H = 20 cph Cups, QQ 1 = 2,000Q 2 = 2,100∆Q = 100 When dealing with a direct relationship based on two performances then the differences between the two performances defining a constant value m ∆Q/∆H = 20 cph and the ratio Q/H = 20 cph

#2 Example There is a Linear relationship between the number of servers, S, and the number of cups sold, Q. Your shop is normally has 20 servers a week and sells 2,000 cups of coffee in a week. When the number of servers is increased to 22, the quantity of coffee sold increases to 2,090 cups.The rate of change at which the café converts the difference in servers into the difference in demand is considered to have a constant value ∆ Q/∆S = 90 cups/2 servers = 45 cups per server When ∆S =2 extra servers are hired the number of coffee sales increases by ∆Q = 90 to a total of 2,090 cups. 1) Because it is a linear relationship the ratio of difference in the cups sold, ∆Q = 90 cups, and ∆S = 2 servers has a constant value of ∆Q/∆H = 90 cups /2 hours = 45 cups per hour The assumption of a linear relationship is reasonable because a café with no servers still sells a some volume of coffee.

A Linear Relationship is one where the ratio of the difference in the two variables has a constant value, ∆Q/∆S = m 0, 0 Number of Servers, S Amount of Product Sold, Q ∆S = ∆Q = 2,090 – 2, servers 22 servers

A Direct Relationship is one where the ratio of the two variables, ∆Q and ∆S has a constant value, ∆Q/∆S = m 0, 0 Change in Number of Servers, ∆S Change in Quantity Sold, ∆Q ∆S = 2 ∆Q = 90 ∆Q/∆S = m = 45 cups per server

#2 Example of Two Performances in a Direct Relationship Defining the Linear Relationship Direct Relationship Performance 1 Direct Relationship Performance 2 Linear relationship ∆ performance = 2-1 Number of Servers, S S 1 = 20S 2 = 22∆S = 2 Q/H = Cups per hour Q 1 /S 1 = 2,000/20 = 100 cps Q 2 /S 2 = 2,090/22 =95 cps ∆Q/∆S = 90/2 ∆Q/∆S = 45 cps Cups, QQ 1 = 2,000Q 2 = 2,090∆Q = 90 When dealing with a direct relationship based on two performances then the differences between the two performances are defining the constant value m ∆Q/∆S = 45 cph The two direct relationships Q/S are not required for the calculation of ∆Q/∆S

#2 Example of Two Performances Defining the Slope of the Linear Relationship Direct Relationship Performance 1 Direct Relationship Performance 2 Linear relationship ∆ performance = 2-1 Number of Servers, S S 1 = 20S 2 = 22∆S = 2 Q/H = Cups per hour NOT Used ∆Q/∆S = 90/2 ∆Q/∆S = 45 cps Cups, QQ 1 = 2,000Q 2 = 2,090∆Q = 90 When dealing with a direct relationship based on two performances then the differences between the two performances are defining the constant value m ∆Q/∆S = 45 cph The two direct relationships Q/S are not required for the calculation of ∆Q/∆S

The End!