Brief Summary
This video provides a step-by-step guide on how to calculate the mean, median, mode, and standard deviation for a given data set. It begins by defining each term and then walks through the calculations with a sample data set. Key steps include arranging the data in ascending order for median calculation, identifying the most frequent number for the mode, and using a formula involving the sum of squared differences from the mean to find the standard deviation.
- Calculating Mean, Median and Mode
- Calculating Standard Deviation
- Variance calculation
The video introduces a statistics problem involving a data set and outlines the task of finding the mean, median, mode, and standard deviation. The given data set is referred to as "x," and the initial instruction is to calculate the mean.
To calculate the mean, sum all the data points and divide by the number of data points. The sum of the data points (24, 31, 12, 38, 12, 15, 12, 28, 16, and 32) is 220. Dividing this sum by the total number of data points, which is 10, gives a mean of 22.
The median is the middle number in a data set arranged in ascending order. The data set is first arranged as follows: 12, 12, 12, 15, 16, 24, 28, 31, 32, and 38. Since there are an even number of data points (10), the median is the average of the two middle numbers, which are 16 and 24. The median is calculated as (16 + 24) / 2 = 20.
The mode is the most frequently occurring number in the data set. In this case, the number 12 appears three times, which is more frequent than any other number. Therefore, the mode is 12.
To calculate the standard deviation, additional columns are created. The first column lists the data points (x) in ascending order. The second column calculates the difference between each data point and the mean (x - x̄). The third column squares the values obtained in the second column ((x - x̄)²).
The mean (x̄) is 22. The values for x - x̄ are calculated as follows:
- 12 - 22 = -10
- 15 - 22 = -7
- 16 - 22 = -6
- 24 - 22 = 2
- 28 - 22 = 6
- 31 - 22 = 9
- 32 - 22 = 10
- 38 - 22 = 16
The values for (x - x̄)² are calculated by squaring the x - x̄ values:
- (-10)² = 100
- (-7)² = 49
- (-6)² = 36
- (2)² = 4
- (6)² = 36
- (9)² = 81
- (10)² = 100
- (16)² = 256
The formula for standard deviation (s) is the square root of the sum of (x - x̄)² divided by (n - 1), where n is the number of data points. The sum of the (x - x̄)² values is 862. Therefore, the standard deviation is calculated as the square root of (862 / (10 - 1)) = √(862 / 9) = √95.778 ≈ 9.79.
The variance is the square of the standard deviation. Using the calculated standard deviation of 9.79, the variance is approximately 9.79 * 9.79 = 95.8. This value is also approximately equal to 95.778, which was the value before taking the square root for the standard deviation.
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Transcript
Full transcript of “How to find the mean, median, mode, standard deviation and variance.”. Click a timestamp to open the video at that moment. Need another video? Use the YouTube transcript generator.
0:01 so we have statistics problems here problem so we have the data set 24 31 12 38 12 15 12 28 16 and we usually call this data data x so we usually um call the row data x so the instruction is to find the mean median mode and the standard deviation so to find the mean so this is equal to the sum of x sum of x so these are the x times n or the number of data points so this is equal to so we have 24 plus 31 plus 12 plus 38 plus 12 plus 15 plus 12 plus 28 plus 16 plus 32 so now this is divided by n or the total number of data so we have one two three four five six seven eight nine ten so we have ten um data data points so this is equal to so we use our calculator so we have 24 plus 31 plus 12 plus 38 plus 12 plus 15 plus 12 plus 28 plus 16 plus 32
2:01 and this is equal to 220 and this is divided by 10 so this is equal to 22. so the mean is equal to 22. so this is the so now to find the median so median so to find the median the median is the number in the middle so we have this and the number in the middle is the median but first since the row data x is not arranged in ascending order so we have to do that first so we have um we have how many 12 1 2 3 so we have 12 12 12 and then we have 15.
3:09 we have 16 and then we have 24. and then 31 32 and then 38 2 4 6 8 10. so we list the um data in ascending order so now the number in the middle is the median so since this is um an even number so the number the two we take the two numbers in the middle so we have so the number in the middle is 16 and 24. so since this is an even number of um data points so we have um 16 plus 24 and we divide it by 2 and this will give us the median so we have 16 plus 24 divide 2 is 20. so the median is 20. so on the mode the mode is the most
4:34 popular number the most popular number so the most popular number is 12 because there are three of them so there are three 12s 115 116 124 128 132 131 and 138 so the most most popular number is 12 so mode is equal to 12. so now to find the standard um deviation we need to make columns so we need so this is x it's easy so i just rewrite the um data here so we have the x and i'm gonna um do them in ascending order so we have 12 12 12 15 16 24 28 31 32 and 38 so now we have another we need another
5:53 column so we have x minus x bar or x bar is also called the mean so this is the mean x bar means mean and then we need x minus x bar squared so now we have 12 so we know that the mean is equal to 22 so we have 12. so this is equal to 12 so x is equal to 12 minus x bar x bar is 22.
6:43 so this is equal to minus 10 so this is minus 10 so this are the same 12 minus 22 is equal to minus 10 so 12 minus 22 equals minus 10 now 15 minus 22 is minus 7 so 15 minus 22 equals minus 7 and 16 minus 22 so 16 minus 22 equals minus 6. now in here so we have 24 minus um 22 so it's equal to 2. [Music] 16 minus 22 is minus 6. so now and then we have 24 minus 22 is 2 so i just write down 2 here this is um 24 minus 22 is equal to 2 so 28 minus 22 is six so this is six then 31 minus 22
8:04 is nine and then 32 minus 22 is 10 and then 38 minus 22 is 16. so this is how you do x minus x bar so 12 minus the mean so now we square x minus x bar this means that minus 10 times minus times minus 10 so we have minus 10 times minus 10 this is what square means so 10 times 10 is a 100 and there are two minus signs so it's positive so basically you just square the number so this is 100 so that means this is also 100 and this is 100 and 7 times 7 is 49 and 6 times 6 is 36 and then 2 times 2
9:02 is 4 and then 6 times 6 is 36 nine times nine is eighty one and then ten times ten is one hundred and sixteen times sixteen is equal to two fifty six so and then we need the um sum so the formula for standard deviation is equal to s is equal to s for standard deviation for sample the sum so the square root of the sum of n of x i use x so x minus x bar squared all over n minus 1. so we need to sum x minus x bar squared so this is the column x minus x bar squared so we need to sum this so we add this so we have 100 plus 100 plus 100 is 300 plus 49 plus 36 plus 4 plus 36 plus 81 plus 100 plus 256 is equal to 862. so now we just plug in into the um formula so the square root of 862 all over n is equal to 10
10:39 we know that so there are 10 data points so 10 minus 1 so get 10 minus one so this is equal to 862 over nine so this is equal to um 95 point seven seven [Music] say eight so now we take the square root of this and it's equal to 9.79 so the standard deviation is equal to 9.79 now the variance is the square of the um standard deviation so variance is equal to the square of standard deviation so times so basically it's this 95 point seven seven eight
11:52 or we can just um 9.79 times 9.79 is you will go back to d it's equal to 95.8 because um i rounded off so it's this number
