Mastering Statistics for GRE Quantitative Reasoning

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. Statistics is one of the segments of GRE quantitative section which is to be structured to adept by means of sensible practices to command over the stipulated ideas, techniques and problems where the base of basic concepts is urged to be trained.

 

• Mean

A ‘mean’ is the average of a set of numbers. One can take a look at the set below, which gives the number of fruits a group of people can eat over the course of time.

Group A: 8
Group B: 9
Group C: 10
Group D: 13

In order to find the mean, one simply need to find the total number of fruits the group as a whole could eat over the whole time i.e.: 8 + 9 + 10 +13 = 40.

Then divide the total by the number of people involved as 40/4=10, which gives us the desired mean. 

• Median
• A ‘median’ is the value of the midpoint of a sequence.
• In GRE test a given set of numbers is not always listed in numerical order. One has to set them rearranged before calculating the median with a view to making the answers corrected.
• When an odd number of items is given one needs to identify the number remains in the middle.  One can consider the example set: {23, 24, 25, 29, 32, 35, 39} where the set denotes us that the middle number is 29, selected as the median.   
• If one adds another number to the end of the above sequence as  {23,24,25,29,32,35,39,48 }

Although 29 and 32 both could be considered to be in the “middle” of this sequence. In order to find the median of a sequence that contains an even number of items, one has to take the average of the two innermost numbers, which will give him/her the median e.g.: here,  

(29 + 32) / 2 = 30.5 is the median of the numbers in the given set.

 

• Mode
• The mode of a set requires one to identify the number that occurs most frequently.
• To easily find the mode, one has to put the numbers in order from least to greatest and count how many times each number occurs. The number that occurs the most is the mode.
• While this seems fairly straightforward, there is an important point to be cautious.

One can consider the set: {29, 23, 21, 25, 23, 28, 26, 30, 28, 28} which can be put in order as

{21, 23, 23, 25, 26, 28, 28, 28, 29, 30} that gives the mode. The mode is 28 as it occurs the most.

• If the set is given as : {29, 23, 21, 25, 23, 26, 30, 28, 28} the first question arisen about this set is how to determine whether 23 or 28 is the mode. In fact, both are the right answer in this case.  There can be more than one mode in a set of numbers.
• In the event that each number only appears once, we can either say that each number is the mode, or that there is no mode in the set at all. 
• Standard deviation
• In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.
• The point of standard deviation is to quantify the standard or average amount by which each piece of data deviates from the average data value in a set. One can consider the two data sets below:

Group I: 33, 40, 58, 64, 81
Group II: 32, 34, 36, 38, 40

• Usually in GRE test, one needs to simply identify which group has the larger standard deviation. One can easily see and realize that the numbers in Group A are far more spread out, meaning that its standard deviation is higher as well.
• The exact standard deviation of a set of numbers is also important remark to be adept at.

The steps to finding the standard deviation of a set of numbers are given orderly: 

1. Firstly one needs to find the mean of the entire set of numbers.
2. Then the differences between the mean of the set and each individual number are to be determined.
3. Then one need to do squaring each of those differences.
4. After that one has to find the average of the new squared values which is known as variance.
5. Finally, the square root of the new squared values will give the standard deviation of the set. So, the standard deviation,  SD = √ Variance

In order to illustrate this, these steps are applied to Group II: {32, 34, 36, 38, 40}

·       The mean of the set of numbers: (32+34+36+38+40) /5 = 36

·       The differences between the mean and each individual number: -4, -2, 0, 2, 4

·       The squares of each difference: 16, 4, 0, 4, 16

·       The mean of these squares: (16+4+0+4+16) /5 = 8. So the variance is 8.

·       The square root of 8 is √8 = 2√2 = 2.828

·       So the standard deviation is 2.828

·       Normal Distribution Curve: The Bell Curve

Figure 1: Bell Curve (Normal Distribution Curve); Source: www.ck12.org

·       It’s important to remember that any Normal Distribution comes with its own the driving parameter which is the standard deviation. 

·       The very center of the Normal Distribution is the mean and median and mode all in one, i.e. in the center point where the standard deviation is zero, Mean=Median=Mode.

·       The standard deviation is applied to measure distances from the mean i.e. the center.  If one goes out a length of one standard deviation (+1SD or -1SD) from the mean on either side, that always includes approximately 68% of the population, a little over two-thirds. This means that on either side, there is 34% of the population, very close to one-third: there’s 34% between the mean and one standard deviation below the mean, and there’s another 34% between the mean and one standard deviation above the mean.  

·       If one goes two standard deviations ((+2SD or -2SD) from the mean in either direction, that always includes approximately 95% of the population. This means that on either side, there is 47.5% of the population.

·       If we go out to three standard deviations from the mean in either direction, that includes 99.7% of the population, with only 0.15% (i.e. 15 people out of 10000) falling in each tail beyond this. 

 

 

Figure 2: Normal Distribution Curve; Source: www.skillstairway.com

 

• Quartiles

Quartiles are the values that divide a list of numbers into quarters:

• Put the list of numbers in order
• Then cut the list into four equal parts
• The Quartiles are at the "cuts"

                                                          

Figure 3: Quartiles (Statistics); Source: www.mathsisfun.com

 

Sometimes a "cut" is between two numbers ... the Quartile is the average of the two numbers.

Figure 4: Quartiles (Statistics); Source: www.mathsisfun.com

 

·      Interquartile Range

The "Interquartile Range" is from Q1 to Q3:  To calculate it just subtract Quartile 1 from Quartile 3, like this:

Figure 5: Interquartile Range (Statistics); Source: www.mathsisfun.com

 

Figure 6: Interquartile Range (Statistics); Source: www.mathsisfun.com

 

• Range

The Range is the difference between the lowest and highest values.

 

Figure 7: Range (Statistics); Source: www.mathsisfun.com

The range can sometimes be misleading when there are extremely high or low values.

Figure 8: Range (Statistics); Source: www.mathsisfun.com

So we may be better off using  Interquartile Range or Standard Deviation.

·      Box and Whisker Plot

We can show all the important values in a "Box and Whisker Plot", like this:

A final example covering everything:

Figure 9: Box Plot (Statistics); Source: www.mathsisfun.com

 

• Percentile

A percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example, the 20th percentile is the value (or score) below which 20% of the observations may be found.

The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests. For example, if a score is at the 86th percentile, where 86 is the percentile rank, it is equal to the value below which 86% of the observations may be found (carefully contrast with in the 86th percentile, which means the score is at or below the value of which 86% of the observations may be found - every score is in the 100th percentile). The 25th percentile is also known as the first quartile (Q₁), the 50th percentile as the median or second quartile (Q₂), and the 75th percentile as the third quartile (Q₃). In general, percentiles and quartiles are specific types of quantiles.

Percentiles represent the area under the normal curve, increasing from left to right. Each standard deviation represents a fixed percentile. Thus, rounding to two decimal places, -3SD (3 Standard Deviation) is the 0.13th percentile -2SD is the 2.28th percentile, -1SD  the 15.87th percentile & 0SD the 50th percentile (both the mean and median of the distribution), +1SD the 84.13th percentile, +2SD the 97.72nd percentile, and +3SD the 99.87th percentile. This is related to the 68–95–99.7 rule or the three-sigma rule.

• Formulas to Remember
• MODE = 3 median – 2 mean
• Quartile deviation, QD = (Q3-Q1)/2  = 5/6  Mean deviation