# Evenly spaced and equally spaced sets.

Evenly and equally spaced sets are the nontrivial topics of GRE and GMAT. Each test aspirants baffle at least once while encountering sets; even and equal spaced sets. There is a fundamental difference between an evenly spaced and an equally spaced set.

## Evenly Spaced set

An evenly spaced set is also known as symmetric distribution. In an evenly spaced set, the data points lie at the same distance from the median. Also, the mean is equal to the median for evenly spaced sets; then for any element larger than the median, there is another element in the set which is equally far below the median, and if you were to remove those two elements, the set which remains would still be symmetric about the original median.

Let’s consider the following examples of evenly spaced sets.

Set A:   {1, 2, 3, 4, 5}

B:   {1, 1, 1, 10, 10, 10}

C:   {1, 5, 5, 5, 9}

All these sets are evenly spaced with the same mean and median. For example, in set A, the mean and median are 3. Each data point on both sides of the mean is fixed or the same; the difference of 2 and 4 from the mean (3) is the same (1).

The same is true for sets B and C. However, the set below is not symmetric;

{1, 1, 5, 5, 9}

### Things to remember for evenly spaced set (symmetric distribution).

1. Mean= median
2. Data points from mean is same or at same distance from mean.
3. Data values can be at fixed distance/difference from each preceding value or can be any pattern such as set B.

In any symmetric set, the median and the mean are equal. As a consequence, the mean and median of any equally spaced set are equal. Further, in any symmetric set, the mean and median are both equal to the average of the smallest and largest elements (or indeed to the average of any ‘symmetric pair’ of elements in the set). Note that the reverse is not true: if the mean and median of a set are equal, that is no guarantee that the set is symmetric. For example, the mean and median of the following set are equal:

{1, 4, 5, 7, 8}

but the set is not symmetric.

## Equally Spaced Set:

GRE and GMAT normally use equally spaced sets. An equally spaced is a subset or part of an evenly spaced set. This type of set is equally spaced if the terms in the set are arranged in increasing order and the difference between consecutive terms is equal to some fixed constant. Consider following examples of equally spaced sets:

{1, 2, 3, 4, 5, 6}

{9.9, 10, 10.1, 10.2}

{110, 180, 250, 320, 390}

In the first example above, the difference between consecutive terms is always equal to 1; in the second example, that difference is equal to 0.1, and in the third example, that difference is equal to 70. We will call the difference between consecutive terms in an equally spaced set the space or spacing of the set. The phrase ‘equally spaced’ is not a phrase you’ll ever see on the GRE & GMAT – it’s just a convenient and simple way to describe a type of set that appears frequently. The technical term used in mathematics to describe an equally spaced set is an ‘arithmetic progression’, or ‘arithmetic sequence’. The GRE & GMAT won’t ever use these phrases either (unless their definition is provided within a question); I only mention them because they appear in some prep books and on internet forums.

Most Common equally spaced sets(Consecutive Integers) Many familiar sets are equally spaced, particularly sets that we encounter in Number Theory. Any set of consecutive integers, consecutive even or odd integers, or consecutive multiples of any integer m is always equally spaced. We even see equally spaced sets when dealing with remainders. If, for example, you were to list, in increasing order, all of the positive integers which give a remainder of 3 when divided by 11, that list would be equally spaced (the spacing would be 11):

3, 14, 25, 36, 47, 58, . . .

### Things to remember for an equally spaced set:

Equally spaced sets are always symmetric, but as can be seen from the examples above, many symmetric sets are not equally spaced.

• All evenly spaced sets are defined if these 3 things are known: 1) The smallest or largest number in the set. 2) the increment. 3) the number of items in the set.
• The Arithmetic Mean (ave.) and median are equal in evenly-spaced sets
• Mean and median of the set are equal to the average of the FIRST and LAST terms in the set.

i.e.Mean = (First + Last) / 2

• Sum of the elements in the set equals the arithmetic mean (ave.) number in the set times the number of items in the set.

e.g. Sum = Average x N (where N is the number of items)

• Number of items in a consecutive set:

For Consecutive Integers: N = (Last – First + 1) where N is the number of items in the set. You have to be inclusive of extremes. E.g. the number of integers between 6 and 10 is not 4, it is 5 {6,7,8,9,10} because both extremes (i.e. 6 and 10) must be counted.

• For Consecutive Multiples: N = {(Last – First)/Increment }+ 1

#### Example

For Example: Set= 2 + 4 + 6+8…………250
Given set is evenly Spaced set with the increment 2
(i) Let’s calculate the Mean of above set : (2 + 250)/2=126
(ii) Number of terms = {2,4, 6, ……250}
= {(250-2)/2} + 1
=124+1
=125
Sum = Mean * number of terms in the given set
126*125
=15,750

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