Variance and Standard Deviation

     This is one of those topics that I genuinely don't remember from my 9th grade Geometry class, or any of my other high school math courses either. I was hoping that looking up the definition in Webster's would help. I learned that variance is the fact or quality of being different, divergent or inconsistent. As a special needs mom, I can totally understand the need for different or divergent parenting, but how does this relate to math?
      The overall theme of the lesson was interpreting data. I do remember from high school that mean is the average number you get from a data set when all points are added together and then divided by the number of points. I also remembered that the median is the middle value of an organized data set, and that the mode was the number seen most often in the data set. So how does variance fit into classifying data with these concepts I do understand? The variance shows me how far each number in the set is from the mean.
The steps to finding variance are as follows
     1. Find the mean
     2. subtract the mean from each data point
     3. Square each difference from step 2
     4. Find the sum of the squares from step 3
     5. divide by the number of data points

This means if i take the teat scores of my friends, 65, 74, 79, 83, 87, and 95, I find the variance by

     1. 65+74+79+83+87+95= 80.5 (mean)
     2. (65-80.5)+(74-80.5)+(79-80.5)+(83-80.5)+(87-80.5)+(95-80.5)
          (-15.5)   +   (-6.5)   +   (-.5)     +       2.5   +    6.5     +      14.5
     3. 240.25 + 42.25 + .25 + 6.25 + 42.25 + 210.25
     4. 541.5
     5. 541.5/6= 90.25

To find how close the data points are to the mean I should do one more step and find the standard deviation. To do this I find the square root of the variance. In the case of my friend's test scores, the standard deviation is 9.5. I interpret this to mean that the vast majority of my friends scores were within 9.5 percentage points of the mean (80.5).

https://mathbitsnotebook.com/Algebra1/StatisticsData/normalgrapha.jpg

I was sure I was going to lose my mind trying to figure this concept out, but my hard work eventually paid off. On a positive note, I now feel fairly confident with standard deviations and the new grey hairs I gained figuring this concept out were easily covered with some cheap box color.