what percentage of the distribution lies within 100 and 200?

Question

The average value of a distribution is 150. The standard deviation is 25

Answer 1

Hi Sheily,

 

Assuming the data are normally distributed, we proceed as follows:

 

Say Y% of the data are less than or equal to 200, and Z% are less than or equal to 100. Then, the percentage of the data between 100 and 200 would simply be Y%-Z%. Here is how to find those percentages:

 

To find the percentage of the data less than or equal to 100, we must first find a z-score pertaining to 100. The z-score is defined as:

 

z = (x - mean)/standard deviation

 

Here, x = 100, the man = 150, and the standard deviation is 25. Then,

 

z₁₀₀ = (100 -150) / 25

 

   = -50/25

 

   = -2

 

We must also find the z score pertaining to 200:

 

z₂₀₀ = (200 - 150) / 25

 

      = 50/25

 

      = 2

 

So, referring to a standard "z table", we look up z₂₀₀ and z₁₀₀. The z table gives 0.9772 for z₂₀₀ = 2, and 0.0228 for z₁₀₀ = -2. These values are the percentage of data less than or equal to 200 and 100, respectively. So, the percentage of data that lies between 100 and 200 is simply 0.9772 - 0.0228 = 0.9544 or 95.44%.