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The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical concept that describes the distribution of data in a population. It states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

To better understand and apply the Empirical Rule, there are several practice problems that can be used. For example, given a set of data with a mean of 50 and a standard deviation of 5, the Empirical Rule can be used to determine the percentage of data that falls within certain ranges. In this case, approximately 68% of the data would fall between 45 and 55, 95% between 40 and 60, and 99.7% between 35 and 65.

Another practice problem could involve using the Empirical Rule to find the probability of a specific outcome. For instance, if a population has a normal distribution with a mean of 100 and a standard deviation of 10, what is the probability that a randomly selected value falls between 90 and 110? By applying the Empirical Rule, we can see that 68% of the data falls within one standard deviation of the mean, so the probability of this occurring would be approximately 68%.

Overall, the Empirical Rule is a useful tool for understanding the distribution of data in a population and can be applied to a variety of practice problems to analyze and interpret data.

Empirical Rule Practice Problems

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The Empirical Rule, sometimes called the 68-95-99.7 rule, states that for a given dataset with a normal distribution:

68% of data values fall within one standard deviation of the mean.

95% of data values fall within two standard deviations of the mean.

99.7% of data values fall within three standard deviations of the mean.

Test your knowledge of the Empirical Rule using the practice problems below.

The height of plants in a certain garden are normally distributed with a mean of 12.3 inches and a standard deviation of 4.1 inches.

Use the Empirical Rule to estimate what percentage of plants are between 8.2 and 16.4 inches tall.

var globalThing= {}; // Globally scoped object

//check if user-entered solution matches correct solution
var user_answer = document.getElementById(‘answer’).value;
if (user_answer == solution) {
document.getElementById(‘output’).innerHTML = “Correct!”
} else {
document.getElementById(‘output’).innerHTML = “Not quite yet…”
}

//toggle answer showing
var result_display = document.getElementById(“words_output”);
result_display.style.display = “block”;
} //end massive check() function

function solution() {
if(globalThing.q_selected==”between”) {
if(globalThing.sd_multiplier==1) {
var solution = 68;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 68% of data values fall within one standard deviation of the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located one standard deviation below the mean and ” + globalThing.sd_above.toFixed(1) + ” is located one standard deviation above the mean.

Thus, 68% of plants are between ” + globalThing.sd_below.toFixed(1) + ” and ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
}
if(globalThing.sd_multiplier==2) {
var solution = 95;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 95% of data values fall within two standard deviations of the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located two standard deviations below the mean and ” + globalThing.sd_above.toFixed(1) + ” is located two standard deviations above the mean.

Thus, 95% of plants are between ” + globalThing.sd_below.toFixed(1) + ” and ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
}
if(globalThing.sd_multiplier==3) {
var solution = 99.7;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 99.7% of data values fall within three standard deviations of the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located three standard deviations below the mean and ” + globalThing.sd_above.toFixed(1) + ” is located three standard deviations above the mean.

Thus, 99.7% of plants are between ” + globalThing.sd_below.toFixed(1) + ” and ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
}
} //end between
if(globalThing.q_selected==”less than”) {
if(globalThing.sd_multiplier==1) {
if(globalThing.sd_selected==globalThing.sd_above) {
var solution = 84;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 68% of data values fall within one standard deviation of the mean. This means that 34% of values fall between the mean and one standard deviation above the mean.

In this example, ” + globalThing.sd_above.toFixed(1) + ” is located one standard deviation above the mean. Since we know that 50% of data values fall below the mean in a normal distribution, a total of 50% + 34% = 84% of values fall below ” + globalThing.sd_above.toFixed(1) + “.

Thus, 84% of plants are less than ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
} else {
var solution = 16; document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 68% of data values fall within one standard deviation of the mean. This means that 34% of values fall between the mean and one standard deviation below the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located one standard deviation below the mean. Since we know that 50% of data values fall above the mean in a normal distribution, a total of 50% + 34% = 84% of values fall above ” + globalThing.sd_below.toFixed(1) + “. This means that 100% – 84% = 16% of values fall below ” + globalThing.sd_below.toFixed(1) + “.

Thus, 16% of plants are less than ” + globalThing.sd_below.toFixed(1) + ” inches tall.”;
}
}
if(globalThing.sd_multiplier==2) {
if(globalThing.sd_selected==globalThing.sd_above) {
var solution = 97.5;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 95% of data values fall within two standard deviations of the mean. This means that 47.5% of values fall between the mean and two standard deviations above the mean.

In this example, ” + globalThing.sd_above.toFixed(1) + ” is located two standard deviations above the mean. Since we know that 50% of data values fall below the mean in a normal distribution, a total of 50% + 47.5% = 97.5% of values fall below ” + globalThing.sd_above.toFixed(1) + “.

Thus, 97.5% of plants are less than ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
} else {
var solution = 2.5;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 95% of data values fall within two standard deviations of the mean. This means that 47.5% of values fall between the mean and two standard deviations below the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located two standard deviations below the mean. Since we know that 50% of data values fall above the mean in a normal distribution, a total of 50% + 47.5% = 97.5% of values fall above ” + globalThing.sd_below.toFixed(1) + “. This means that 100% – 97.5% = 2.5% of values fall below ” + globalThing.sd_below.toFixed(1) + “.

Thus, 2.5% of plants are less than ” + globalThing.sd_below.toFixed(1) + ” inches tall.”;
}
}
if(globalThing.sd_multiplier==3) {
if(globalThing.sd_selected==globalThing.sd_above) {
var solution = 99.85;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 99.7% of data values fall within three standard deviations of the mean. This means that 49.85% of values fall between the mean and three standard deviations above the mean.

In this example, ” + globalThing.sd_above.toFixed(1) + ” is located three standard deviations above the mean. Since we know that 50% of data values fall below the mean in a normal distribution, a total of 50% + 49.85% = 99.85% of values fall below ” + globalThing.sd_above.toFixed(1) + “.

Thus, 99.85% of plants are less than ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
} else {
var solution = 0.15;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 99.7% of data values fall within three standard deviations of the mean. This means that 49.85% of values fall between the mean and three standard deviations below the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located three standard deviations below the mean. Since we know that 50% of data values fall above the mean in a normal distribution, a total of 50% + 49.85% = 99.85% of values fall above ” + globalThing.sd_below.toFixed(1) + “. This means that 100% – 99.85% = 0.15% of values fall below ” + globalThing.sd_below.toFixed(1) + “.

Thus, 0.15% of plants are less than ” + globalThing.sd_below.toFixed(1) + ” inches tall.”;
}
}
} //end less than
if(globalThing.q_selected==”greater than”) {
if(globalThing.sd_multiplier==1) {
if(globalThing.sd_selected==globalThing.sd_above) {
var solution = 16;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 68% of data values fall within one standard deviation of the mean. This means that 34% of values fall between the mean and one standard deviation above the mean.

In this example, ” + globalThing.sd_above.toFixed(1) + ” is located one standard deviation above the mean. Since we know that 50% of data values fall below the mean in a normal distribution, a total of 50% + 34% = 84% of values fall below ” + globalThing.sd_above.toFixed(1) + “. This means that 100% – 84% = 16% of values fall above ” + globalThing.sd_above.toFixed(1) + “.

Thus, 16% of plants are greater than ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
} else {
var solution = 84;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 68% of data values fall within one standard deviation of the mean. This means that 34% of values fall between the mean and one standard deviation below the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located one standard deviation below the mean. Since we know that 50% of data values fall above the mean in a normal distribution, a total of 50% + 34% = 84% of values fall above ” + globalThing.sd_below.toFixed(1) + “.

Thus, 84% of plants are greater than ” + globalThing.sd_below.toFixed(1) + ” inches tall.”;
}
}
if(globalThing.sd_multiplier==2) {
if(globalThing.sd_selected==globalThing.sd_above) {
var solution = 2.5;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 95% of data values fall within two standard deviations of the mean. This means that 47.5% of values fall between the mean and two standard deviations above the mean.

In this example, ” + globalThing.sd_above.toFixed(1) + ” is located two standard deviations above the mean. Since we know that 50% of data values fall below the mean in a normal distribution, a total of 50% + 47.5% = 97.5% of values fall below ” + globalThing.sd_above.toFixed(1) + “. This means that 100% – 97.5% = 2.5% of values fall above ” + globalThing.sd_above.toFixed(1) + “.

Thus, 2.5% of plants are greater than ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
} else {
var solution = 97.5;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 95% of data values fall within two standard deviations of the mean. This means that 47.5% of values fall between the mean and two standard deviations below the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located two standard deviations below the mean. Since we know that 50% of data values fall above the mean in a normal distribution, a total of 50% + 47.5% = 97.5% of values fall above ” + globalThing.sd_below.toFixed(1) + “.

Thus, 97.5% of plants are greater than ” + globalThing.sd_below.toFixed(1) + ” inches tall.”;
}
}
if(globalThing.sd_multiplier==3) {
if(globalThing.sd_selected==globalThing.sd_above) {
var solution = 0.15;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 99.7% of data values fall within three standard deviations of the mean. This means that 49.85% of values fall between the mean and three standard deviations above the mean.

In this example, ” + globalThing.sd_above.toFixed(1) + ” is located three standard deviations above the mean. Since we know that 50% of data values fall below the mean in a normal distribution, a total of 50% + 49.85% = 99.85% of values fall below ” + globalThing.sd_above.toFixed(1) + “. This means that 100% – 99.85% = 0.15% of values fall above ” + globalThing.sd_above.toFixed(1) + “.

Thus, 0.15% of plants are greater than ” + globalThing.sd_above.toFixed(1) + ” inches tall.”;
} else {
var solution = 99.85;
document.getElementById(‘solution_words’).innerHTML = “The Empirical Rule states that for a given dataset with a normal distribution, 99.7% of data values fall within three standard deviations of the mean. This means that 49.85% of values fall between the mean and three standard deviation below the mean.

In this example, ” + globalThing.sd_below.toFixed(1) + ” is located three standard deviations below the mean. Since we know that 50% of data values fall above the mean in a normal distribution, a total of 50% + 49.85% = 99.85% of values fall above ” + globalThing.sd_below.toFixed(1) + “.

Thus, 99.85% of plants are greater than ” + globalThing.sd_below.toFixed(1) + ” inches tall.”;
}
}
} //end greater than

//toggle hide/show solution
var solution_div = document.getElementById(“solution_div”);
solution_div.style.display = “block”;
} //end massive solution() function

function gen() {
var mean = Math.round(jStat.uniform.sample(20, 50)*10)/10;
var sd = Math.round(jStat.uniform.sample(2, 6)*10)/10;

var sd_options = [1, 2, 3];
globalThing.sd_multiplier = sd_options[Math.floor(Math.random()*sd_options .length)];

globalThing.sd_above = mean – (-globalThing.sd_multiplier*sd);
globalThing.sd_below = mean – (globalThing.sd_multiplier*sd);

sd_above_below = [globalThing.sd_above, globalThing.sd_below];
globalThing.sd_selected = sd_above_below[Math.floor(Math.random()*sd_above_below.length)];

var q_options = [“between”, “less than”, “greater than”];
globalThing.q_selected = q_options[Math.floor(Math.random()*q_options .length)];

if (globalThing.q_selected == “less than”) {
document.getElementById(‘scenario’).innerHTML = “less than ” + globalThing.sd_selected.toFixed(1);
} else if (globalThing.q_selected == “greater than”) {
document.getElementById(‘scenario’).innerHTML = “greater than ” + globalThing.sd_selected.toFixed(1);
} else {
document.getElementById(‘scenario’).innerHTML = “between ” + globalThing.sd_below.toFixed(1) + ” and ” + globalThing.sd_above.toFixed(1);
}

//fill in mean and sd in initial question
document.getElementById(‘mean’).innerHTML = mean;
document.getElementById(‘sd’).innerHTML = sd;

//toggle answer & solution to hide and clear input field
var result_display = document.getElementById(“words_output”);
result_display.style.display = “none”;
var solution_div = document.getElementById(“solution_div”);
solution_div.style.display = “none”;
document.getElementById(‘answer’).value = “”;
} //end massive gen() function

//generate initial question
gen();

What are some practice problems that illustrate the Empirical Rule?

Empirical Rule Practice Problems

Requst a

Scale

Related terms:

Requst a

Scale