**Related SAS Tutorials**

- 5A – (3:01) Numeric Measures using PROC MEANS

**Related SPSS Tutorials**

- 5A – (8:00) Numeric Measures using EXPLORE

So far we have learned about different ways to quantify the center of a distribution. A measure of center by itself is not enough, though, to describe a distribution.

Consider the following two distributions of exam scores. Both distributions are centered at 70 (the median of both distributions is approximately 70), but the distributions are quite different.

The first distribution has a much larger variability in scores compared to the second one.

In order to describe the distribution, we therefore need to supplement the graphical display not only with a measure of center, but also with a measure of the variability (or spread) of the distribution.

In this section, we will discuss the three most commonly used measures of spread:

- Range
- Inter-quartile range (IQR)
- Standard deviation

Although the **measures of center** did approach the question differently, they do **attempt to measure the same point in the distribution** and thus are comparable.

However, the three **measures of spread** provide very different ways to quantify the variability of the distribution and **do not try to estimate the same quantity**.

In fact, the three **measures of spread** **provide information about three different aspects of the spread** of the distribution which, together, give a more complete picture of the spread of the distribution.

The **range** covered by the data is the most intuitive measure of variability. The range is exactly the distance between the smallest data point (min) and the largest one (Max).

- Range = Max – min

**Note: **When we first looked at the histogram, and tried to get a first feel for the spread of the data, we were actually approximating the range, rather than calculating the exact range.

Here we have the Best Actress Oscar winners’ data

34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61 21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33

In this example:

- min = 21 (Marlee Matlin for
*Children of a Lesser God*, 1986) - Max = 80 (Jessica Tandy for
*Driving Miss Daisy*, 1989)

The range covered by all the data is 80 – 21 = 59 years.

While the range quantifies the variability by looking at the range covered by ALL the data,

the **Inter-Quartile Range** or** IQR** measures the variability of a distribution by giving us the range covered by the MIDDLE 50% of the data.

**IQR**= Q3 – Q1**Q3**= 3^{rd}Quartile = 75^{th}Percentile**Q1**= 1^{st}Quartile = 25^{th}Percentile

The following picture illustrates this idea: (Think about the horizontal line as the data ranging from the min to the Max). **IMPORTANT NOTE:** **The “lines” in the following illustrations are not to scale. The equal distances indicate equal amounts of data NOT equal distance between the numeric values.**

Although we will use software to calculate the quartiles and IQR, we will illustrate the basic process to help you fully understand.

To calculate the IQR:

- Arrange the data in increasing order, and find the median M. Recall that the median divides the data, so that 50% of the data points are below the median, and 50% of the data points are above the median.

- Find the median of the lower 50% of the data. This is called the first quartile of the distribution, and the point is denoted by Q1. Note from the picture that Q1 divides the lower 50% of the data into two halves, containing 25% of the data points in each half. Q1 is called the first quartile, since one quarter of the data points fall below it.
- Repeat this again for the top 50% of the data. Find the median of the top 50% of the data. This point is called the third quartile of the distribution, and is denoted by Q3.

Note from the picture that Q3 divides the top 50% of the data into two halves, with 25% of the data points in each.Q3 is called the third quartile, since three quarters of the data points fall below it. - The middle 50% of the data falls between Q1 and Q3, and therefore: IQR = Q3 – Q1

**Comments:**

- The last picture shows that Q1, M, and Q3 divide the data into four quarters with 25% of the data points in each, where the median is essentially the second quartile. The use of IQR = Q3 – Q1 as a measure of spread is therefore particularly appropriate when the median M is used as a measure of center.

- We can define a bit more precisely what is considered the bottom or top 50% of the data. The bottom (top) 50% of the data is all the observations whose position in the ordered list is to the left (right) of the location of the overall median M. The following picture will visually illustrate this for the simple cases of n = 7 and n = 8.

Note that when n is **odd** (as in n = 7 above), the median is **not** included in either the bottom or top half of the data; When n is **even** (as in n = 8 above), the data are naturally divided into two halves.

To find the IQR of the Best Actress Oscar winners’ distribution, it will be convenient to use the stemplot.

Q1 is the median of the bottom half of the data. Since there are 16 observations in that half, Q1 is the mean of the 8th and 9th ranked observations in that half:

Q1 = (31 + 33) / 2 = 32

Similarly, Q3 is the median of the top half of the data, and since there are 16 observations in that half, Q3 is the mean of the 8th and 9th ranked observations in that half:

Q3 = (41 + 42) / 2 = 41.5

IQR = 41.5 – 32 = 9.5

Note that in this example, the range covered by all the ages is 59 years, while the range covered by the middle 50% of the ages is only 9.5 years. While the whole dataset is spread over a range of 59 years, the middle 50% of the data is packed into only 9.5 years. Looking again at the histogram will illustrate this:

**Comment:**

- Software packages use different formulas to calculate the quartiles Q1 and Q3. This should not worry you, as long as you understand the idea behind these concepts. For example, here are the quartile values provided by three different software packages for the age of best actress Oscar winners:

**R:**

**Minitab:**

**Excel:**

Q1 and Q3 as reported by the various software packages differ from each other and are also slightly different from the ones we found here. This should not worry you.

There are different acceptable ways to find the median and the quartiles. These can give different results occasionally, especially for datasets where n (the number of observations) is fairly small.

As long as you know what the numbers mean, and how to interpret them in context, it doesn’t really matter much what method you use to find them, since the differences are negligible.

So far, we have introduced two measures of spread; the range (covered by all the data) and the inter-quartile range (IQR), which looks at the range covered by the middle 50% of the distribution. We also noted that the IQR should be paired as a measure of spread with the median as a measure of center.

We now move on to another measure of spread, the **standard deviation**, which quantifies the spread of a distribution in a completely different way.

The idea behind the standard deviation is to quantify the spread of a distribution by measuring how far the observations are from their mean. The standard deviation gives the average (or typical distance) between a data point and the mean.

There are many notations for the standard deviation: SD, s, Sd, StDev. Here, we’ll use **SD** as an abbreviation for standard deviation, and use s as the symbol.

The **sample standard deviation formula** is:

where,

s = sample standard deviation

n = number of scores in sample

= sum of…

and

= sample mean

In order to get a better understanding of the standard deviation, it would be useful to see an example of how it is calculated. In practice, we will use a computer to do the calculation.

The following are the number of customers who entered a video store in 8 consecutive hours:

7, 9, 5, 13, 3, 11, 15, 9

To find the standard deviation of the number of hourly customers:

**Find the mean, x-bar, of your data:**

(7 + 9 + 5 + 13 + 3 + 11 + 15 + 9)/8 = 9

**Find the deviations from the mean:**

- The differences between each observation and the mean here are

(7 – 9), (9 – 9), (5 – 9), (13 – 9), (3 – 9), (11 – 9), (15 – 9), (9 – 9)

-2, 0, -4, 4, -6, 2, 6, 0

- Since the standard deviation attempts to measure the average (typical) distance between the data points and their mean, it would make sense to average the deviation we obtained.
**Note,**however**, that the sum of the deviations is zero.**- This is always the case, and is the reason why we need a more complex calculation.

**To solve the previous problem, in our calculation, we square each of the deviations.**

(-2)^{2}, (0)^{2}, (-4)^{2}, (4)^{2}, (-6)^{2}, (2)^{2}, (6)^{2}, (0)^{2}

4, 0, 16, 16, 36, 4, 36, 0

**Sum the squared deviations and divide by***n*– 1:

(4 + 0 + 16 + 16 + 36 + 4 + 36 + 0)/(8 – 1)

(112)/(7) = 16

- The reason we divide by
*n*-1 will be discussed later. - This value, the sum of the squared deviations divided by n – 1, is called the
**variance**. However, the variance is not used as a measure of spread directly as the units are the square of the units of the original data.

**The standard deviationof the data is the square root of the variance calculated in step 4:**

- In this case, we have the square root of 16 which is 4. We will use the lower case letter
*s*to represent the standard deviation.

*s* = 4

- We take the square root to obtain a measure which is in the original units of the data. The units of the variance of 16 are in “squared customers” which is difficult to interpret.
- The units of the standard deviation are in “customers” which makes this measure of variation more useful in practice than the variance.

Recall that the average of the number of customers who enter the store in an hour is 9.

**The interpretation of the standard deviation is that on average, the actual number of customers who enter the store each hour is 4 away from 9.**

**Comment: **The importance of the numerical figure that we found in #4 above called the variance (=16 in our example) will be discussed much later in the course when we get to the inference part.

- It should be clear from the discussion thus far that the SD should be paired as a measure of spread with the mean as a measure of center.

- Note that the only way, mathematically, in which the SD = 0, is when all the observations have the same value (Ex: 5, 5, 5, … , 5), in which case, the deviations from the mean (which is also 5) are all 0. This is intuitive, since if all the data points have the same value, we have no variability (spread) in the data, and expect the measure of spread (like the SD) to be 0. Indeed, in this case, not only is the SD equal to 0, but the range and the IQR are also equal to 0. Do you understand why?

- Like the mean, the SD is strongly influenced by outliers in the data. Consider the example concerning video store customers: 3, 5, 7, 9, 9, 11, 13, 15 (data ordered). If the largest observation was wrongly recorded as 150, then the average would jump up to 25.9, and the standard deviation would jump up to SD = 50.3. Note that in this simple example, it is easy to see that while the standard deviation is strongly influenced by outliers, the IQR is not. The IQR would be the same in both cases, since, like the median, the calculation of the quartiles depends only on the order of the data rather than the actual values.

The last comment leads to the following very important conclusion:

- Use the
**mean and the standard deviation**as measures of center and spread for**reasonably symmetric distributions with no extreme outliers.**

**For all other cases**, use**the five-number summary = min, Q1, Median, Q3, Max**(which gives the median, and easy access to the IQR and range). We will discuss the five-number summary in the next section in more detail.

- The
**range**covered by the data is the most intuitive measure of spread and is exactly the distance between the smallest data point (min) and the largest one (Max).

- Another measure of spread is the
**inter-quartile range (IQR)**, which is the range covered by the middle 50% of the data.

- IQR = Q3 – Q1, the difference between the third and first quartiles.
- The
**first quartile (Q1)**is the value such that one quarter (25%) of the data points fall below it, or the median of the bottom half of the data. - The
**third quartile (Q3)**is the value such that three quarters (75%) of the data points fall below it, or the median of the top half of the data.

- The

- The
**IQR**is generally used as a measure of spread of a distribution when the**median**is used as a measure of center.

- The
**standard deviation**measures the spread by reporting**a typical (average) distance between the data points and their mean.**

- It is appropriate to use the
**standard deviation**as a measure of spread with the**mean**as the measure of center.

- Since the
**mean and standard deviations are highly influenced by extreme observations**, they should be used as numerical descriptions of the center and spread**only for distributions that are roughly symmetric, and have no extreme outliers. In all other situations, we prefer the 5-number summary.**

**Related SAS Tutorials**

- 5A – (3:01) Numeric Measures using PROC MEANS

**Related SPSS Tutorials**

- 5A – (8:00) Numeric Measures using EXPLORE

Intuitively speaking, a numerical measure of center describes a “typical value” of the distribution.

The two main numerical measures for the center of a distribution are the **mean** and the **median**.

In this unit on Exploratory Data Analysis, we will be calculating these results based upon a **sample** and so we will often emphasize that the values calculated are the **sample mean** and **sample median**.

Each one of these measures is based on a completely different idea of describing the center of a distribution.

We will first present each one of the measures, and then compare their properties.

The **mean** is the **average** of a set of observations (i.e., the sum of the observations divided by the number of observations).

The **mean** is the **average** of a set of observations

- The sum of the observations divided by the number of observations).
- If the n observations are written as

- their mean can be written mathematically as:their mean is:

We read the symbol as “x-bar.” The bar notation is commonly used to represent the **sample mean**, i.e. the mean of the sample.

Using any appropriate letter to represent the variable (x, y, etc.), we can indicate the sample mean of this variable by adding a bar over the variable notation.

We will continue with the Best Actress Oscar winners example (Link to the Best Actress Oscar Winners data).

34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61 21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33

The mean age of the 32 actresses is:

We add all of the ages to get **1233** and **divide** **by** the number of ages which was **32** to get **38.5. **

We denote this result as **x-bar** and called the **sample mean**.

Note that the sample mean gives a measure of center which is higher than our approximation of the center from looking at the histogram (which was 35). The reason for this will be clear soon.

Often we have large sets of data and use a frequency table to display the data more efficiently.

Data were collected from the last three World Cup soccer tournaments. A total of 192 games were played. The table below lists the number of goals scored per game (not including any goals scored in shootouts).

Total # Goals/Game | Frequency |
---|---|

0 | 17 |

1 | 45 |

2 | 51 |

3 | 37 |

4 | 25 |

5 | 11 |

6 | 3 |

7 | 2 |

8 | 1 |

**To find the mean** number of goals scored per game, we would need to **find the sum of all 192 numbers, and then divide that sum by 192.**

Rather than add 192 numbers, we use the fact that the same numbers appear many times. For example, the number 0 appears 17 times, the number 1 appears 45 times, the number 2 appears 51 times, etc.

If we add up 17 zeros, we get 0. If we add up 45 ones, we get 45. If we add up 51 twos, we get 102. Repeated addition is multiplication.

Thus, the **sum of the 192 numbers**

** = 0(17) + 1(45) + 2(51) + 3(37) + 4(25) + 5(11) + 6(3) + 7(2) + 8(1) = 453.**

The **sample mean** is then **453 / 192 = 2.359**.

Note that, in this example, the values of 1, 2, and 3 are the most common and our average falls in this range representing the bulk of the data.

The **median** M is the midpoint of the distribution. It is the number such that half of the observations fall above, and half fall below.

To find the median:

- Order the data from smallest to largest.

- Consider whether n, the number of observations, is even or odd.
- If n is
**odd**, the median M is the center observation in the ordered list. This observation is the one “sitting” in the**(n + 1) / 2**spot in the ordered list. - If n is
**even**, the median M is the**mean**of the**two center observations**in the ordered list. These two observations are the ones “sitting” in the**(n / 2)**and**(n / 2) + 1**spots in the ordered list.

- If n is

For a simple visualization of the location of the median, consider the following two simple cases of n = 7 and n = 8 ordered observations, with each observation represented by a solid circle:

**Comments:**

- In the images above, the dots are equally spaced, this need not indicate the data values are actually equally spaced as we are only interested in listing them in order.

- In fact, in the above pictures, two subsequent dots could have exactly the same value.

- It is clear that the value of the median will be in the same position regardless of the distance between data values.

To find the median age of the Best Actress Oscar winners, we first need to order the data.

It would be useful, then, to use the stemplot, a diagram in which the data are already ordered.

- Here n = 32 (an even number), so the median M, will be the mean of the two center observations
- These are located at the (n / 2) = 32 / 2 =
**16th**and (n / 2) + 1 = (32 / 2) + 1 =**17th**

Counting from the top, we find that:

- the 16th ranked observation is 35
- the 17th ranked observation also happens to be 35

Therefore, the median M = (35 + 35) / 2 = 35

As we have seen, the **mean** and the **median**, the most common **measures of center**, each describe the center of a distribution of values in a different way.

- The mean describes the center as an average value, in which the
**actual values**of the data points play an important role. - The median, on the other hand, locates the middle value as the center, and the
**order**of the data is the key.

To get a deeper understanding of the differences between these two measures of center, consider the following example. Here are two datasets:

Data set A → 64 65 66 68 70 71 73 |

Data set B → 64 65 66 68 70 71 730 |

For dataset A, the mean is 68.1, and the median is 68.

Looking at dataset B, notice that all of the observations except the last one are close together. The observation 730 is very large, and is certainly an outlier.

In this case, the median is still 68, but the mean will be influenced by the high outlier, and shifted up to 162.

The message that we should take from this example is:

Therefore:

- For symmetric distributions with no outliers: the mean is approximately equal to the median.

- For skewed right distributions and/or datasets with high outliers: the mean is greater than the median.

- For skewed left distributions and/or datasets with low outliers: the mean is less than the median.

**Conclusions… When to use which measures?**

- Use the sample mean as a measure of center for symmetric distributions with no outliers.
- Otherwise, the median will be a more appropriate measure of the center of our data.

- The two main numerical measures for the center of a distribution are the mean and the median. The mean is the average value, while the median is the middle value.

- The mean is very sensitive to outliers (as it factors in their magnitude), while the median is resistant to outliers.

- The mean is an appropriate measure of center for symmetric distributions with no outliers. In all other cases, the median is often a better measure of the center of the distribution.