**Statistics**

Statistics is the study of the patterns and relationships of numbers and data. There are four main concepts that may appear on the test:

**Median**

When a set of numbers is arranged in order of size, the *median* is the middle number. For example, the median of the set {8, 9, 10, 11, 12} is 10 because it is the middle number. In this case, the median is also the mean (average). But this is usually not the case. For example, the median of the set {8, 9, 10, 11, 17} is 10 because it is the middle number, but the mean is 11 = $\frac{\mathrm{8+9+10+11+17}}{5}$ . If a set contains an even number of elements, then the median is the average of the two middle elements. For example, the median of the set {1, 5, 8, 20} is 6.5 (= $\frac{5+8}{2}$ ) .

**Example:**

What is the median of 0, â€“2, 256 , 18, âˆš2? Arranging the numbers from smallest to largest (we could also arrange the numbers from the largest to smallest; the answer would be the same), we get â€“2, 0, âˆš2, 18, 256. The median is the middle number, âˆš2.

**Mode**

The mode is the number or numbers that appear most frequently in a set. Note that this definition allows a set of numbers to have more than one mode.

**Example 1:**

What is the mode of 3, â€“4, 3 , 7, 9, 7.5 ?

The number 3 is the mode because it is the only number that is listed more than once.

**Example 2:**

What is the mode of 2, Ï€, 2 , â€“9, Ï€, 5 ?

Both 2 and Ï€ are modes because each occurs twice, which is the greatest number of occurrences for any number in the list.

**Range**

The range is the distance between the smallest and largest numbers in a set. To calculate the range, merely subtract the smallest number from the largest number.

**Example:**

What is the range of 2, 8, 1 , â€“6, Ï€, 1/2 ?

The largest number in this set is 8, and the smallest number is â€“6. Hence, the range is 8 â€“ (â€“6) = 8 + 6 = 14.

**Standard Deviation**

On the test, you are not expected to know the definition of standard deviation. However, you may be presented with the definition of standard deviation and then be asked a question based on the definition. To make sure we cover all possible bases, weâ€™ll briefly discuss this concept.

*Standard deviation* measures how far the numbers in a set vary from the setâ€™s mean. If the numbers are scattered far from the setâ€™s mean, then the standard deviation is large. If the numbers are bunched up near the setâ€™s mean, then the standard deviation is small.

**Example:**

Which of the following sets has the larger standard deviation?

A = {1, 2, 3, 4, 5}

B = {1, 4, 15, 21, 34}

All the numbers in Set A are within 2 units of the mean, 3. All the numbers in Set B are greater than 5 units from the mean, 15 (except, of course, the mean itself). Hence, the standard deviation of Set B is greater.

**Standard Deviation**

**Standard Deviation**

On the test, you are not expected to know the definition of standard deviation. However, you may be presented with the definition of standard deviation and then be asked a question based on the definition. To make sure we cover all possible bases, weâ€™ll briefly discuss this concept.

*Standard deviation* measures how far the numbers in a set vary from the setâ€™s mean. If the numbers are scattered far from the setâ€™s mean, then the standard deviation is large. If the numbers are bunched up near the setâ€™s mean, then the standard deviation is small.

**Example:** Which of the following sets has the larger standard deviation?

A = {1, 2, 3, 4, 5}

B = {1, 4, 15, 21, 34}

All the numbers in Set A are within 2 units of the mean, 3. All the numbers in Set B are greater than 5 units from the mean, 15 (except, of course, the mean itself). Hence, the standard deviation of Set B is greater.