2.2 - One Quantitative Variable

2.2 - One Quantitative Variable

One quantitative variable is covered in Sections 2.2 and 2.3 of the Lock5 textbook. In these sections, you will learn how to describe the distribution of a quantitative variable in terms of shape, central tendency, and variability. You will be introduced to the normal distribution, scores, percentiles, graphs, and the five-number-summary.


2.2.1 - Graphs: Dotplots and Histograms

2.2.1 - Graphs: Dotplots and Histograms

Dotplots and histograms are both graphical displays that can be used with one quantitative variable. In both of these plots the horizontal axis represents the values of the variable. The number of dots in a dotplot, or the height of the bars in a histogram, represent the number of cases with each value or range of values. 

Dotplot

Dotplot of Height (inches)

Histogram

Histogram of Height (inches)

MinitabExpress  – Dotplot

To create a dotplot in Minitab Express:

  1. Open the data set:
  2. On a PC or Mac: Select GRAPHS > Dotplot
  3. Select Simple
  4. Double click the variable Verbal SAT (2005) in the box on the left to insert the variable into the Y variable box
  5. Click OK

This should result in the following dotplot:

Dotplot of Verbal SAT (2005)

Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.

MinitabExpress  – Histogram

To create a histogram in Minitab Express:

  1. Open the data set:
  2. On a PC or Mac: Select GRAPHS > Histogram
  3. Select Simple
  4. Double click the variable Verbal SAT(2005) in the box on the left to insert the variable into the Y variable box
  5. Click OK

This should result in the following histogram:

Histogram of Verbal SAT (2005)

Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.


2.2.2 - Outliers

2.2.2 - Outliers

Some observations within a set of data may fall outside the general scope of the other observations. Such observations are called outliers. Outliers can be identified by looking at a dotplot or histogram. In Lesson 3 you'll learn about boxplots which can also be used to identify outliers. When constructing a boxplot, Minitab Express identifies outliers using mathematical methods that you will see next week. This week we will identify outliers by making a relatively subjective judgement from a given a list of data points, a dotplot, or a histogram.

Example: Dotplot of Hours Watching TV

Watching Television

A sample of STAT 200 students was surveyed and asked how many hours per week they watch television. A dotplot was constructed using these data.

Dotplot of Hours Per Week Watching TV

The right-most dot is definitely an outlier because it is much higher than any other points. The other higher points, around 55, 50, and 46, may be outliers. Next week we will learn about some mathematical methods for identifying outliers that can help us make decisions in cases like this where it is not obvious which values are outliers. 

Example: Histogram of Best Marriage Age

This sample of students was also asked what they believed was the best age to get married. A histogram was constructed using these data.

Histogram of Best Marriage Age

There appear to be three outliers in this sample, all on the higher end.


2.2.3 - Shape

2.2.3 - Shape

Quantitative variables are often discussed in terms of their shape. Both dotplots and histograms can be used to interpret a distribution's shape. A distribution may be described in terms of symmetry and skewness.

Symmetrical Distribution

A distribution that is similar on both sides of the center.

Symmetrical Distribution

Normal Distribution

One specific type of symmetrical distribution. This is also known as a bell-shaped distribution. 

Normal Distribution

Skewed
A distribution in which values are more spread out on one side of the center than on the other.
Right Skewed

A distribution in which the higher values (towards the right on a number line) are more spread out than the lower values. This is also known as positively skewed.

Right Skewed Distribution

Left Skewed

A distribution in which the lower values (towards the left on a number line) are more spread out than the higher values. This is also known as negatively skewed.

Left Skewed Distribution


2.2.4 - Measures of Central Tendency

2.2.4 - Measures of Central Tendency

Quantitative variables are often summarized using numbers to communicate their central tendency. The mean, median, and mode are three of the most commonly used measures of central tendency.

Mean

The numerical average; calculated as the sum of all of the data values divided by the number of values.

The sample mean is represented as \(\overline{x}\) ("x-bar") and the population mean is denoted as the Greek letter \(\mu\) ("mu"). The formula is the same for the sample mean and the population mean.

Population Mean
\(\mu=\dfrac{\Sigma x}{N}\)
Sample Mean
\(\overline {x} = \dfrac{\Sigma x}{n}\)
Median
The middle of the distribution that has been ordered from smallest to largest; for distributions with an even number of values, this is the mean of the two middle values.
Mode
The most frequently occurring value(s) in the distribution, may be used with quantitative or categorical variables.

Example: Hours Spent Studying

Studying

A professor asks a sample of 7 students how many hours they spent studying for the final. Their responses are: 5, 7, 8, 9, 9, 11, and 13.

Mean

\(\overline{x} = \dfrac{\sum x}{n} =\dfrac{5+7+8+9+9+11+13}{7} =\dfrac{62}{7} =8.857\)

The mean is 8.857 hours.

Median

The observations are already in order from smallest to largest. The middle observation is 9 hours. The median is 9 hours. 

Mode

The most frequently occurring observation was 9 hours. The mode is 9 hours.

In this example, the mean, median, and mode are all similar. Recall from our discussion of shape, the mean, median, and mode are all equal when a distribution is symmetric. This distribution of hours spent studying is probably close to symmetrical.

Example: Test Scores

A teacher wants to examine students’ test scores. Their scores are: 74, 88, 78, 90, 94, 90, 84, 90, 98, and 80.

Mean

\(\overline{x}\: =\: \dfrac{\sum x}{n} = \dfrac{74+88+78+90+94+90+84+90+98+80}{10} = \dfrac{866}{10}=86.6\)

The mean score was 86.6.

Median

First, we need to put the scores in order from lowest to highest: 74, 78, 80, 84, 88, 90, 90, 90, 94, 98

Because there is an even number of scores, the median will be the mean of the middle two values. The middle two values are 88 and 90. \(\frac{88+90}{2}=89\)

The median is 89. 

Mode

The most frequently occurring score was 90. There were 3 students who scored a 90; this is the mode. Because this distribution has one mode, it is unimodal.

In this example the mean is slightly lower than the median which is slightly lower than the mode. Recall from our discussion of shape that this occurs when a distribution is skewed to the left. This distribution is probably slightly skewed to the left.

Example: Household Size

A group of children are asked how many people live in their household. The following data is collected: 4, 3, 6, 2, 2, 4, 3.

Mean

\(\overline{x} = \dfrac{\sum x}{n}=\dfrac{4+3+6+2+2+4+3}{7}=\dfrac{24}{7}=3.429\)

The mean household size in this group of children is 3.429 people.

Median

First, we need to put all of the values in order from smallest to largest: 2, 2, 3, 3, 4, 4, 6

The value in the middle of this distribution is 3. The median is 3. 

Mode

In this distribution, the most common values are 2, 3, and 4. Each of these values occurs twice. There are 3 modes: 2, 3, and 4. This distribution is multimodal.


2.2.4.1 - Skewness & Central Tendency

2.2.4.1 - Skewness & Central Tendency

The preferred measure of central tendency often depends on the shape of the distribution. In a symmetrical distribution, the mean, median, and mode are all equal. In these cases, the mean is often the preferred measure of central tendency. 

Symmetrical: Mean = Median = Mode

For distributions that are strongly skewed or have outliers, the median is often the most appropriate measure of central tendency because in skewed distributions the mean is pulled out toward the tail. The median is more resistant to outliers compared to the mean. Of these three measures of central tendency, the mean is most influenced by outliers. Below you will see how the direction of skewness impacts the order of the mean, median, and mode.

Skewed to the left: Mean < Median < Mode

Skewed to the right: Mode < Median < Mean


2.2.5 - Measures of Spread

2.2.5 - Measures of Spread

Variance and standard deviation are measures of variability. The standard deviation is the most commonly used measure of variability when data are quantitative and approximately normally distributed. When computing the standard deviation by hand, it is necessary to first compute the variance. The standard deviation is equal to the square root of the variance. Here, you will learn how to compute these values by hand. After this lesson, you will always be computing standard deviation using software such as Minitab Express.

Standard Deviation
Roughly the average difference between individual data values and the mean. The standard deviation of a sample is denoted as \(s\). The standard deviation of a population is denoted as \(\sigma\).
Sample Standard Deviation
\(s=\sqrt{\dfrac{\sum (x-\overline{x})^{2}}{n-1}}\)

In order to compute the standard deviation for a sample we first compute deviations. The sum of the squared deviations (SS) divided by \(n-1\), this is the variance (\(s^2\)).

The square root of the variance is the standard deviation: \(\sqrt{s^2}=s\).

Deviation
An individual score minus the mean.
Sum of Squared Deviations
Deviations squared and added together. This is also known as the sum of squares or SS.
Variance
Approximately the average of all of the squared deviations; for a sample represented as \(s^{2}\).
Sum of Squares
\(SS={\sum (x-\overline{x})^{2}}\)
Sample Variance
\(s^{2}=\dfrac{\sum (x-\overline{x})^{2}}{n-1}\)

There are a number of methods for calculating the standard deviation. If you look through different textbooks or search online, you may find different formulas and procedures. To compute the standard deviation for a sample, we will use the formulas above and the following steps:

Step 1: Compute the sample mean: \(\overline{x} = \frac{\sum x}{n}\).

Step 2: Subtract the sample mean from each individual value: \(x-\overline{x}\), these are the deviations.

Step 3: Square each deviation: \((x-\overline{x})^{2}\), these are the squared deviations.

Step 4: Add the squared deviations: \(\sum (x-\overline{x})^{2}\), this is the sum of squares.

Step 5: Divide the sum of squares by \(n-1\): \(\frac{\sum (x-\overline{x})^{2}}{n-1}\), this is the sample variance \((s^{2})\).

Step 6: Take the square root of the sample variance: \(\sqrt{\frac{\sum (x-\overline{x})^{2}}{n-1}}\), this is the sample standard deviation.

Video Example

The video below walks through an example of computing a sample standard deviation by hand.

 

Example: Hours Spent Studying

Studying

A professor asks a sample of 7 students how many hours they spent studying for the final. Their responses are: 5, 7, 8, 9, 9, 11, and 13.

Step 1: Compute the mean

\(\overline{x} = \dfrac{\sum x}{n}=\dfrac{5+7+8+9+9+11+13}{7}=8.857\)

Step 2: Compute the deviations

\(x\) \(x - \overline{x}\)
5 \(5 - 8.857 = -3.857\)
7 \(7 - 8.857 = -1.857\)
8 \(8 - 8.857 = -0.857\)
9 \(9 - 8.857 = 0.143\)
9 \(9 - 8.857 = 0.143\)
11 \(11 - 8.857 = 2.143\)
13 \(13 - 8.857 = 4.143\)

Step 3: Square the deviations

\(x\) \(x - \overline{x}\) \((x-\overline{x})^{2}\)
5 \(5 - 8.857 = -3.857\) \(-3.857^{2} = 14.876\)
7 \(7 - 8.857 = -1.857\) \(-1.857^{2} = 3.448\)
8 \(8 - 8.857 = -0.857\) \(-0.857^{2} = 0.734\)
9 \(9 - 8.857 = 0.143\) \(0.143^{2} = 0.0020\)
9 \(9 - 8.857 = 0.143\) \(0.143^{2} = 0.0020\)
11 \(11 - 8.857 = 2.143\) \(2.143^{2} = 4.592\)
13 \(13 - 8.857 = 4.143\) \(4.143^{2} = 17.164\)

Step 4: Sum the squared deviations

\(SS=\sum (x-\overline{x})^{2}=14.876+3.448+0.734+.020+.020+4.592+17.164=40.854\)

The sum of squares is 40.854

Step 5: Divide by n - 1 to compute the variance

\(s^{2}=\dfrac{\sum (x-\overline{x})^{2}}{n-1}=\dfrac{40.854}{7-1}=6.809\)

The variance is 6.809

Step 6: Take the square root of the variance

\(s=\sqrt{s^{2}}=\sqrt{6.809}=2.609\)

The standard deviation is 2.609


2.2.6 - Minitab Express: Central Tendency & Variability

2.2.6 - Minitab Express: Central Tendency & Variability

Minitab Express may be used to compute descriptive statistics such as the mean, median, mode, standard deviation, and variance. 

Note that these are the default setting in Minitab Express:

Minitab Express: Descriptive Statistics Defaults

If you want the mode or variance, you will need to select them under the Statistics tab.

MinitabExpress  – Central Tendency

To obtain measures of central tendency and variability in Minitab Express:

  1. Open the data set:
  2. On a PC: from the menu select STATISTICS > Describe 
    On a Mac: from the menu select Statistics > Summary Statistics > Descriptive Statistics
  3. Double click the variable Height in the box on the left to insert the variable into the Variable box
  4. Click on the Statistics tab and select the descriptive statistics that you want displayed
  5. Click OK

This should result in the following output:

Descriptive Statistics: Height
Statistics
Variable N N* Mean SE Mean StDev Minimum Q1 Median Maximum
Height 525 0 67.0090 0.1947 4.4616 51.0000 64.0000 67.0000 82.0000
Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.


2.2.7 - The Empirical Rule

2.2.7 - The Empirical Rule

A normal distribution is symmetrical and bell-shaped. 

Normal Distribution

The Empirical Rule is a statement about normal distributions. Your textbook uses an abbreviated form of this, known as the 95% Rule, because 95% is the most commonly used interval. The 95% Rule states that approximately 95% of observations fall within two standard deviations of the mean on a normal distribution.

Normal Distribution
A specific type of symmetrical distribution, also known as a bell-shaped distribution
Empirical Rule

On a normal distribution about 68% of data will be within one standard deviation of the mean, about 95% will be within two standard deviations of the mean, and about 99.7% will be within three standard deviations of the mean

The normal curve showing the empirical rule.
mean−2s mean−1s mean+1s mean−3s mean+3s mean mean+2s 68% 95% 99.7%
95% Rule
On a normal distribution approximately 95% of data will fall within two standard deviations of the mean; this is an abbreviated form of the Empirical Rule

Example: Pulse Rates

Checking Pulse

Suppose the pulse rates of 200 college men are bell-shaped with a mean of 72 and standard deviation of 6.

  • About 68% of the men have pulse rates in the interval \(72\pm1(6)=[66, 78]\).
  • About 95% of the men have pulse rates in the interval \(72\pm2(6)=[60, 84]\).
  • About 99.7% of the men have pulse rates in the interval \(72\pm 3(6)=[54, 90]\).

Example: IQ Scores

 

IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

  • About 68% of individuals have IQ scores in the interval \(100\pm 1(15)=[85,115]\).
  • About 95% of individuals have IQ scores in the interval \(100\pm 2(15)=[70,130]\).
  • About 99.7% of individuals have IQ scores in the interval \(100\pm 3(15)=[55,145]\).

2.2.8 - z-scores

2.2.8 - z-scores

Often we want to describe an observation in relation to the distribution of all observations. We can do this using a z-score. By converting observations to z-scores, we can compare observations from different distributions. 

z-score

Distance between an individual score and the mean in standard deviation units; also known as a standardized score.

z-score
\(z=\dfrac{x - \overline{x}}{s}\)

\(x\) = original data value
\(\overline{x}\) = mean of the original distribution
\(s\) = standard deviation of the original distribution

This equation could also be rewritten in terms of population values: \(z=\frac{x-\mu}{\sigma}\)

Later in the course, we will learn more about the z-distribution, which is a special case of the normal distribution. 

z-distribution

A bell-shaped distribution with a mean of 0 and standard deviation of 1, also known as the standard normal distribution.

Distribution Plot

Example: Milk

Dairy Cow

A study of 66,831 dairy cows found that the mean milk yield was 12.5 kg per milking with a standard deviation of 4.3 kg per milking (data from Berry, et al., 2013).

A cow produces 18.1 kg per milking. What is this cow’s z-score?

\(z=\frac{x-\overline{x}}{s} =\frac{18.1-12.5}{4.3}=1.302\)

This cow’s z-score is 1.302; her milk production was 1.302 standard deviations above the mean.

 

A cow produces 12.5 kg per milking. What is this cow’s z-score?

\(z=\frac{x-\overline{x}}{s} =\frac{12.5-12.5}{4.3}=0\)

This cow’s z-score is 0; her milk production was the same as the mean.

 

A cow produces 8 kg per milking. What is this cow’s z-score?

\(z=\frac{x-\overline{x}}{s} =\frac{8-12.5}{4.3}=-1.047\)

This cow’s z-score is -1.047; her milk production was 1.047 standard deviations below the mean.

 

Berry, D. P., Coyne, J., Boughlan, B., Burke, M., McCarthy, J., Enright, B., Cromie, A. R., McParland, S. (2013). Genetics of milking characteristics in dairy cows. Animal, 7(11), 1750-1758.

Example: Comparing Test Scores

SAT-Math scores are normally distributed with a mean of 500 and standard deviation of 100. ACT-Math scores are normally distributed with a mean of 18 and standard deviation of 6. A student has taken both tests. They scored 600 on the SAT-Math and 22 on the ACT-Math. Which score is more impressive?

We can't directly compare the student's SAT and ACT scores because they are on different scales. We can convert these test scores into z-scores so we can directly compare them.

\(z_{SAT}=\frac{600-500}{100}=1\)

This student scored 1 standard deviation above the mean on the SAT-Math.

\(z_{ACT}=\frac{22-18}{6}=0.667\)

This student scored 0.667 standard deviations above the mean on the ACT-Math.

The student's SAT-Math score is more impressive than their ACT-Math score because the z-score is higher. They scored better than a larger proportion of other test takers on the SAT-Math.

Practice: Computing z-scores

Type in the answer you think is correct - then click the 'Check' button to see how you did.

Click the right arrow to proceed to the next question.  When you have completed all of the questions you will see how many you got right and the correct answers.

For each question, compute the z-score.

Type in the answer you think is correct - then click the 'Check' button to see how you did.

Click the right arrow to proceed to the next question.  When you have completed all of the questions you will see how many you got right and the correct answers.

For each question, compute the z-score.


2.2.9 - Percentiles

2.2.9 - Percentiles

There are slightly different definitions of percentiles and different statistical software and textbooks may use different formulas. In this course, we will be using the definition from your textbook: 

Percentile
Proportion of a distribution less than a given value.

Example: Test Scores

Test scores are often reported in terms of percentiles. For example, if a student scores in the 90th percentile on a test then he or she scored better than 90% of students who took the test.


2.2.10 - Five Number Summary

2.2.10 - Five Number Summary
Five Number Summary
Minimum, Q1, Median, Q3, Maximum

Q1 is the first quartile, this is the 25th percentile
Q3 is the third quartile, this is the 75th percentile

Five number summaries are used to describe some of the key features of a distribution. Using the values in a five number summary we can also compute the range and interquartile range.

Range
The difference between the maximum and minimum values.
Range
\(Range = Maximum - Minimum\)
Note:
The range is heavily influenced by outliers. For this reason, the interquartile range is often preferred because it is resistant to outliers.
Interquartile range (IQR)
The difference between the first and third quartiles.
Interquartile Range
\(IQR = Q_3 - Q_1\)

Example: Hours Spent Studying

Studying

A professor asks a sample of students how many hours they spent studying for the final. The five number summary for their responses is (5, 7, 9, 11, 13).

Range

The maximum is 13 and the minimum is 5.

\(Range = 13 - 5 = 8\)

Interquartile Range

The third quartile is 11 and the first quartile is 7.

\(IQR = Q_3 - Q_1 = 11 - 7 = 4\)

Example: Test Scores

A teacher wants to examine students’ test scores. The five number summary for their scores is (74, 80, 89, 90, 98).

Range

The highest score is 98. The lowest score is 74.

\(Range = 98 - 74 = 24\)

Interquartile Range

The third quartile is 90 and the first quartile is 80.

\(IQR = Q3 - Q1 = 90 - 80 = 10\)


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