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 = $\frac{8+9+10+11+17}{5}$ . 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

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.