Common terms in Statistics I

Content

• Mean
• Mode
• Median
• Quartiles
• Variance
• Standard Deviation

Here is our sample data, x = [2, 6, 8, 1, 56, 13, 8, -5, 4, 6, 23].\ Length, n = 11.

Mean

Average of the data. Add all items, divide by length to calculate mean. For x, mean = 11 (rounded)

Mode

Most common item. For x, mode = 6.

Median

The median is the middle number in a data set. Steps to find the median-

• Sort the data from low to high. For x, the sorted array is: -5, 1, 2, 4, 6, 6, 8, 13, 23, 56
• If n is odd, find the middle item. If n is even, find the middle two item and calculate their mean

Here, since n is 11, 6th item is the median = 6.

Quartile

Quartiles are values that divide the data in 4 regions. The regions are known as the lowest 25% of numbers, next lowest 25% of numbers(up to median), second highest 25% of numbers (above median), the highest 25% of numbers. So to have 4 region, we need 3 points called as Q1,Q2,Q3. In layman’s term, the Q1 is greater than or equal to the lowest 25% of the number and so on. To calculate quartiles, data have to be sorted.

1st Quartile

$$Q_{1}=\left(\frac{n+1}{4}\right)^{th}term$$

For x, Q1 = 2

2nd Quartile

It’s the same as median which is 6.

3rd Quartile

$$Q_{3}=\left(\frac{3}{4}(n+1)\right)^{th}term$$

For x, Q3 = 13

Variance

It explains how far the data is spread out from their mean. Calculated as the average of the squared difference from the mean. For x, do $(2-11)^{2}+ (6-11)^{2} + .... + (23-11)^{2}$ and finally divide the sum by 11. Result is, 247.90

Standard Deviation (SD)

This is simply the square root of variance. Explains if a number is normal or not, a number can be big or small compared to the other items of the dataset. SD tells us how tightly the data is clustered around the mean. A small SD indicates that the data is tightly clustered. A large SD tells that the data is more spread apart.

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