The type of information on how scores are spread from the center of a distribution is obtained by

• Introduction
• Range
• Inter-Quartile Range (IQR)
• Standard Deviation
• Properties of the Standard Deviation
• Choosing Numerical Measures
• Let’s Summarize

CO-4: Distinguish among different measurement scales, choose the appropriate descriptive and inferential statistical methods based on these distinctions, and interpret the results.

LO 4.4: Using appropriate graphical displays and/or numerical measures, describe the distribution of a quantitative variable in context: a) describe the overall pattern, b) describe striking deviations from the pattern

LO 4.7: Define and describe the features of the distribution of one quantitative variable (shape, center, spread, outliers).

Introduction

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.

Range

LO 4.11: Define and calculate the range of one quantitative variable.

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.

EXAMPLE: Best Actress Oscar Winners

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.

Inter-Quartile Range (IQR)

LO 4.12: Define and calculate Q1, Q3, and the IQR for one quantitative variable

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 = 3rd Quartile = 75th Percentile
• Q1 = 1st Quartile = 25th 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:

1. 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.
   
   
2. 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.
   
3. 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.
   
4. The middle 50% of the data falls between Q1 and Q3, and therefore: IQR = Q3 – Q1
   

Comments:

1. 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.
   

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2. 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.

EXAMPLE: Best Actress Oscar Winners

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.

Standard Deviation

LO 4.13: Define and calculate the standard deviation and variance of one quantitative variable.

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.

Idea

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.

Notation

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.

Formula

The sample standard deviation formula is:

where,

s = sample standard deviation

n = number of scores in sample

and

Calculation

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.

EXAMPLE: Video Store Customers

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:

1. Find the mean, x-bar, of your data:

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

1. 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.

1. 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

1. 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.

1. 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 sto 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.

Properties of the Standard Deviation

1. 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.
   

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2. 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?
   

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3. 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:

Choosing Numerical Measures

LO 4.10: Choose the appropriate measures for a quantitative variable based upon the shape of the distribution.

• 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.

Let’s Summarize

• 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 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.