Mean when a Number is Multiplied to each term

Transcript

Mean when a Number is Multiplied to each termLet’s Consider the data: 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5. We found mean as Mean = (8 + 3 + 10 + 13 + 4 + 6 + 7 + 7 + 8 + 8 + 5)/11 = 7.18 (approx.) Now, let’s multiply each term with 5 So, our new data becomes 40, 15, 50, 65, 20, 30, 35, 35, 40, 40, 30 Now, New Mean = (𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑡𝑒𝑟𝑚𝑠)/(𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠) = (40 + 15 + 50 + 65 + 20 + 30 + 35 + 35 + 40 + 40 + 25)/11 = 395/11 = 35.91 (approx.) We notice that New Mean = Old mean × 5 Checking on a dot plot Let’s try to prove it generally General Statement Let the original values be 𝑥_1,𝑥_2,…,𝑥_𝑛 and their average be 𝑎. 𝒂=(𝒙_𝟏 + 𝒙_𝟐 + ⋯ + 𝒙_𝒏)/𝒏 If we multiply 5 to eachvalue, the new average calculation looks like this: " New Average "=((5𝑥_1 )+(5𝑥_2 )+(5𝑥_3 )+⋯+(5𝑥_𝑛 ))/𝑛 Since 5 is multiplied to each number, we can take it out =((𝑥_1 + 𝑥_2 + ⋯ + 𝑥_𝑛 ) × 5)/𝑛 =(𝑥_1 + 𝑥_2 + ⋯ + 𝑥_𝑛)/𝑛 × 5 Substitute our original average 𝑎 back in =𝒂 × 𝟐 Thus, we can write If number c is multiplied to each term New Mean = Old mean × c If number d is divided to each term New Mean = (𝑶𝒍𝒅 𝒎𝒆𝒂𝒏)/𝒅