Interpreting Standard Deviation: Videos & Practice Problems

The standard deviation measures data spread, with a large value indicating wide dispersion and a small value indicating close clustering. The empirical rule states that in a normal distribution, approximately 68% of data lies within one standard deviation of the mean, 95% within two, and 99.7% within three. Values two or more standard deviations from the mean are considered significant. For example, if the mean weight of milk bottles is 12 ounces with a standard deviation of 0.5 ounces, a weight of 11 ounces is significant as it falls two standard deviations below the mean.

Empirical Rule of Standard Deviation and Range Rule of Thumb

Understanding standard deviation is crucial for interpreting datasets, as it provides insights into how data points relate to the overall distribution. A large standard deviation indicates that data points are widely spread out, while a small standard deviation suggests they are closely clustered around the mean. Two important concepts that help in analyzing datasets using standard deviation are the empirical rule and the range rule of thumb.

The empirical rule, also known as the 68-95-99.7 rule, applies specifically to datasets that follow a normal distribution, which is visually represented by a bell curve. This rule estimates the percentage of data that falls within certain intervals around the mean. For a normal distribution:

• Approximately 68% of the data lies within one standard deviation (σ) of the mean (μ), specifically between μ - σ and μ + σ.
• About 95% of the data falls within two standard deviations, between μ - 2σ and μ + 2σ.
• Approximately 99.7% of the data is found within three standard deviations, between μ - 3σ and μ + 3σ.

For example, if the mean weight of milk bottles is 12 ounces with a standard deviation of 0.5 ounces, the interval from 10.5 ounces to 13.5 ounces represents three standard deviations from the mean. According to the empirical rule, 99.7% of the milk bottles will fall within this range, indicating that the company can confidently assert that their products meet this weight specification.

The range rule of thumb builds on the empirical rule by identifying values that are significantly different from the mean. It states that any value that lies two or more standard deviations away from the mean is considered significant. This means that values falling below μ - 2σ or above μ + 2σ are noteworthy because they are much higher or lower than expected. In the previous example, if we consider whether a weight of 11 ounces is significant, we find that it is exactly two standard deviations below the mean (12 - 0.5 - 0.5). Therefore, 11 ounces is classified as a significant value.

In summary, the empirical rule provides a framework for understanding the distribution of data in a normal dataset, while the range rule of thumb helps identify outliers that may warrant further investigation. Together, these tools enhance our ability to analyze and interpret data effectively.

A sample of 500 random adult books in a library has an average of 312 pages with a standard deviation of 26 pages. Find the percentage of books in the sample with less than 338 pages using the Empirical Rule of Standard Deviation.

• A. $84\%$
• B. $34\%$
• C. $50\%$
• D. $16\%$

Answer: (A)

A sample of 500 random adult books in a library has an average of 312 pages with a standard deviation of 26 pages. According to the Empirical Rule of Standard Deviation, find the central range of page lengths containing 95% of the books in the sample.

• A. $(312,364)$
• B. $(260,312)$
• C. $(260,364)$
• D. $(286,338)$

Answer: (C)

The average birth weight at a hospital is 6.5lbs. with a standard deviation of 1.4lbs. What is the lowest weight which would be considered significantly high?

• A. $2.8$
• B. $1.4$
• C. $9.3$
• D. $3.7$
• E. $6.5$

Answer: (C)

Empirical Rule of Standard Deviation and Range Rule of Thumb Example 1

The empirical rule, also known as the 68-95-99.7 rule, is a statistical principle that helps us understand the distribution of data in a normal distribution. In this context, we analyze a sample of 250 wait times at a restaurant, which has a mean wait time of 8 minutes and a standard deviation of 2.5 minutes. This rule states that approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations.

To find the percentage of wait times between 3 minutes and 10.5 minutes, we first determine how many standard deviations these values are from the mean. The lower bound of 3 minutes is calculated as the mean (8 minutes) minus two standard deviations (5 minutes), while the upper bound of 10.5 minutes is the mean plus one standard deviation (2.5 minutes). By adding the percentages of the areas under the normal curve corresponding to these bounds, we find that 81.5% of the wait times fall within this interval.

Next, we explore the percentage of wait times greater than 0.5 minutes (30 seconds). This value is three standard deviations below the mean (8 minutes). Since 50% of the data lies above the mean, we can calculate the additional percentages from the areas between the mean and three standard deviations below. Adding these percentages gives us a total of 99.85%, indicating that nearly all wait times exceed 30 seconds.

Finally, to determine the minimum wait time required to receive a coupon, we focus on the highest 2.5% of wait times. Since 95% of the data lies within two standard deviations of the mean, the highest 2.5% corresponds to values above this range. Calculating this, we find that the minimum wait time for a coupon is 13 minutes, which is two standard deviations above the mean (8 minutes + 5 minutes).

In summary, the empirical rule provides a powerful tool for understanding the distribution of wait times, allowing us to make informed decisions based on statistical analysis.

Empirical Rule of Standard Deviation and Range Rule of Thumb Example 2

In this example, we explore the application of the range rule of thumb to estimate the standard deviation of song lengths based on a sample of 200 randomly selected songs. The typical duration of these songs ranges from 2.3 minutes to 4.7 minutes. To apply the range rule of thumb, we recognize that any value within two standard deviations of the mean is considered not significant.

To find the standard deviation, we first identify the lower and upper bounds of the song lengths: 2.3 minutes (lower bound) and 4.7 minutes (upper bound). The range of these values can be calculated as:

$(4.7-2.3)$ = 2.4 \text{ minutes}

This total range of 2.4 minutes corresponds to four standard deviations (from the lower bound to the upper bound). To find the value of one standard deviation, we divide the total range by four:

$2.4/4=0.6$ \text{ minutes}

Thus, the estimated standard deviation of the song lengths is 0.6 minutes. This application of the range rule of thumb provides a useful method for estimating variability in data sets, particularly when direct measurements of standard deviation are not available.

Here’s what students ask on this topic:

What does the empirical rule of standard deviation tell us about data distribution?

The empirical rule of standard deviation, also known as the 68-95-99.7 rule, applies to datasets with a normal distribution (bell curve). It states that approximately 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps estimate the spread of data and identify intervals where most data points are likely to fall. For example, if the mean weight of milk bottles is 12 ounces and the standard deviation is 0.5 ounces, then 68% of the weights will be between 11.5 and 12.5 ounces, 95% between 11 and 13 ounces, and 99.7% between 10.5 and 13.5 ounces.

How can you determine if a value is significant using the range rule of thumb?

The range rule of thumb identifies significant values in a dataset by checking if they are two or more standard deviations away from the mean. Values within two standard deviations of the mean are considered typical, while those outside this range are significant. For example, if the mean is 12 and the standard deviation is 0.5, a value of 11 is significant because it is two standard deviations below the mean (12 - 2 × 0.5 = 11). Similarly, a value of 13 would be significant as it is two standard deviations above the mean (12 + 2 × 0.5 = 13).

What percentage of data lies within three standard deviations of the mean in a normal distribution?

In a normal distribution, approximately 99.7% of the data lies within three standard deviations of the mean. This interval includes most of the dataset, leaving only 0.3% of the data outside this range (0.15% on each tail). For example, if the mean weight of milk bottles is 12 ounces and the standard deviation is 0.5 ounces, then 99.7% of the weights will fall between 10.5 ounces (12 - 3 × 0.5) and 13.5 ounces (12 + 3 × 0.5).

Why is the empirical rule important in interpreting standard deviation?

The empirical rule is crucial for interpreting standard deviation because it provides a clear framework for understanding data distribution in a normal dataset. It helps estimate the percentage of data within specific intervals around the mean, making it easier to identify typical values and outliers. For example, knowing that 95% of data lies within two standard deviations of the mean allows researchers to focus on the remaining 5% as potentially significant or unusual values. This rule is widely used in statistics, quality control, and research to analyze and interpret data effectively.

How does the shape of the dataset affect the application of the empirical rule?

The empirical rule applies only to datasets with a normal distribution, characterized by a bell-shaped curve that is symmetrical about the mean. If the dataset is skewed or does not follow a normal distribution, the empirical rule may not accurately describe the data spread. For example, in a normal distribution, 68% of data lies within one standard deviation of the mean, but this percentage may differ in a skewed dataset. Therefore, verifying the dataset's shape is essential before using the empirical rule.