Statistics 101: Proportions, ratios and rates

Catalogue number: 892000062021003

Release date: May 3, 2021

This video is intended for viewers who wish to gain a basic understanding of three types of measures: proportions, ratios, and rates. No previous knowledge is required. Although proportions, ratios and rates are similar to each other, they each have different properties.

In this video, you will learn the differences between these three measures. In addition, by the end of the video, you will be able to answer the following question: How is each one calculated, and when is it best to use one rather than the other?

Data journey step
Analyze, model
Data competency
  • Data exploration
  • Data interpretation
Audience
Basic
Suggested prerequisites
N/A
Length
13:16
Cost
Free

Watch the video

Statistics 101: Proportions, ratios and rates - Transcript

Statistics 101: Proportions, ratios and rates - Transcript

(The Statistics Canada symbol and Canada wordmark appear on screen with the title: "Statistics 101 Proportions, ratios, and rates")

Statistics 101: Proportions, ratios, and rates

Often the easiest way to tell a data story is to simply compare one given number with another. When we hear "75 percent of people think this or two thirds of companies do that", what we are hearing are the results of analyzing a given set of data, and three helpful measures for doing so are proportions, ratios and rates. All of these measures are easy to calculate, use and understand. This video will introduce you to the concepts of proportions, ratios and rates and the differences between them.

Learning goals

This video is intended for viewers who wish to gain a basic understanding of three types of measures: proportions, ratios and rates. No previous knowledge is required. Although proportions, ratios and rates are similar to each other, they each have different properties. In this video, you will learn the differences between these three measures. In addition, by the end of this video, you will be able to answer the following questions. How is each one calculated, and when is it best to use one rather than the other?

Steps of a data journey

(Text on screen: Supported by a foundation of stewardship, metadata, standards and quality)

(Diagram of the Steps of the data journey: Step 1 - Find, gather, protect; Step 2 - explore, clean, describe; Step 3 - analyze, model; Step 4 - tell the story. The data journey is supported by a foundation of stewardship, metadata, standards and quality.)

This diagram is a visual representation of the data journey, from collecting the data to exploring, cleaning, describing and understanding the data, to analyzing the data and lastly, to communicating with others the story the data tell.

Step 2: Explore, clean and describe

(Diagram of the Steps of the data journey with an emphasis on Step 2 - explore, clean, describe.)

In the data journey, the use of proportions, ratios and rates are part of the explore, clean and describe step and are also used to Analyze and Model.

What is a proportion?

Let's start with proportions. A proportion is a part, share or a number considered in comparative relation to a whole. The smallest value for a proportion is zero or the largest possible value is one. A proportion can be expressed as a percentage by multiplying its value by one hundred. Proportions are useful when you want to compare a number to a total. For example, in an audience of 50 people, five are left handed. This can be expressed as a proportion by dividing five by fifty, for a result of 0.10 to ten percent by multiplying 0.1 by 100.

Proportions: An example

Imagine you have a standard deck of 52 playing cards. The deck contains 13 cards of each suit: diamonds, hearts, clubs and spades. For the sake of this example, we will assume that there are no jokers or any extra cards. What is the proportion of diamonds in the deck? To calculate this proportion, we should first count the number of diamond cards. There are 13. Then, we would divide this number by the total number of cards in the deck, which is 52. This gives us a proportion of 0.25. Expressed as a percentage, the answer is 25 percent.

What is a ratio?

Now, let's move on to ratios. A ratio tells us the relative size of two values. The difference between proportion and a ratio is that with ratios, you have more freedom to compare. The notation is also different. While ratios can be expressed as numbers or percentages, they are most commonly expressed with a colon. Placing a colon in between the numbers two and one for example, should be read as "a ratio of two to one". This means that the first value is twice as large as the second value. A ratio of three to two, meanwhile, means that, for every three of the first item, there are two of the second item. A good example of this can be observed if you ever tried to cook rice. Depending on the rice, the instructions might say "two parts water to one part rice", and what that means is that it doesn't matter if you're trying to cook rice for 2 people or 20 people, just knowing the ratio of 2:1 water to rice means that whatever volume of rice you have, double it to get the amount of water and you have the right amounts to start cooking.

Another difference between proportion and ratio, is that any ratio, you can choose which quantity is used as a reference. Instead of counting two parts of water for every part of rice, you could choose to express it as one part of rice for every two parts of water. It is often easier to express the biggest quantity as a relative size of the smallest, but in some context, the alternative can be preferable. Finally, the ratio is often expressed. In the simplest terms. A ratio of 4:2 is better understood if it is being expressed as 2:1.

Ratios: an example

Let's return to our playing card example. What is the ratio of diamonds to hearts? To calculate this ratio, we should first count the number of diamonds. From a previous question, we know that this number is 13. Next, let's count the second part of our ratio, which is the number of hearts. The number of hearts is equal to the number of diamonds, which is again 13. Expressed as a ratio. We have 13 to 13. However, this ratio can be simplified by dividing both sides by the same number. Here, both sides can be divided by 13, which gives us a ratio of one to one. This means that, for every diamond card in the deck, there is one heart card.

Knowledge check

Let's see if you can calculate proportions and ratios. First, what proportions of the deck is made up of queen's? Pause the video here and restart once you think you have the answer.

4 out of the 52 cards are queens. The proportion of queens in a deck is about 0.08 or 8%.

Knowledge check

Now, what is the ratio of face cards to non face cards in the deck? Pause the video now to work it out and restart once you are ready to see the full answer.

There are 4 suits, diamonds, spades, clubs and hearts. In each suit, there are 3 different types of face cards, jacks, queens and kings and 10 different types of non face cards. 4 times 3 is 12, so there are 12 face cards in the deck.4 times 10 is 40, so there are 40 non-face cards in the deck. Meaning, the ratio of face cards to non-face cards is 12:40, or more simply, for every 3 face cards, there are 10 non-face cards.

What is a rate?

Let's move on to rates. The simplest definition of a rate is that it is one quantity divided by another quantity. With that definition, both proportions and ratios could qualify as being rates. So what is the difference between rates and the other two measures? Proportions and ratios are commonly used to compare quantities that have the same unit of measurement. In the deck of cards for example, both quantities that are compared are counts representing number of cards. In a rate, the two quantities that are compared often have different units of measurements. For instance, speed is a rate: it is the distance traveled (in kilometers for instance) divided by the time it took to travel that distance (in hours). The rate would have kilometers per hour as a unit of measurement.

Calculating rates: Growth rate

(Histogram demonstrating the amount in savings account ($) on January 1st, 2018, 2019 and 2020. Respectively, the amounts are $800; $1,200 & $900)

One type of rate that is particularly useful is a growth rate. A growth rate compares the change in a measurement over a period of time to the values of the measurement at the start of the period. It is very useful to assess change over time. This figure represents the amount of money available in a savings account at the start of the years 2018, 2019 and 2020. Suppose you wanted to evaluate the change in the savings account during the year 2018. What would you do?

To calculate the growth rate of the amount of money in your savings account, first you would calculate the change that occurred during the year. You do this by taking the amount of money saved at the start of 2019, which was $1,200 , and subtracting the amount that was in the account at the start of 2018, which was $800. The difference is $400. Then, you divide this difference by the amount in the account at the start of 2018. In this case this means you would divide $400 by $800. The result is 0.5. This growth rate can then be multiplied by 100 if you want to express it as a percentage. This gives you 50%. Meaning, during 2018, the amount in your savings account increased by half the value of what you contained at the beginning of the year.

Knowledge check

(Histogram demonstrating the amount in savings account ($) on January 1st, 2018, 2019 and 2020. Respectively, the amounts are $800; $1,200 & $900)

Now it's your turn. What was the growth rate of the amount in the savings account during year 2019, that is, between January 1st, 2019 and January 1st, 2020? Pause the video now and restart when you are ready to see the full answer.

The answer is negative 0.25 or -25%, and here's why. First, we took the amount of money in the account at the start of 2020, which was $900, and subtracted the amount from the start of 2019, which was $1,200. This gave us a negative number of -$300. Then we divide that difference by the amount of the start of 2019, $1,200. Which resulted in a growth rate of -0.25 or -25%. The negative growth rate means that the amount of money in the savings account decreased in 2019. One quarter, or 25% of the initial value was lost during the year.

Comparison of proportions, ratios and rates

(Table containing the definitions of each 3 measure. The columns, from left to right, are titled as follows: Measure | Description | Conditions | Notations | Examples in Official Statistics. From the first to last line: Proportion | Part, share or number considered in comparative relation to a whole | 0, 1, or any value between 0 and 1 | Number or percentage | Proportion of Canadian population living BC; Ratio | The relative sizes of two values | Positive numbers (any value) | A:B (a ratio of A to B) or a number | Gender wage ratio; Rate | One quantity divided by another | None (negative values are allowed) | Number or percentage | Annual population growth rate)

Let's review the three measures we've covered. A proportion is a part, share or number considered in comparative relation to a whole. It can be equal to 0, 1 or any value between 0 and 1. It can be expressed as a number or percentage. One example in official statistics would be the proportion of the Canadian population who lives in a given province. A ratio is the relative size of two values. It can be used to compare two parts to another. Ratios can be made up of any positive values and are commonly expressed as two numbers separated by colon, or by a single number. An example in official statistics is the gender wage ratio, which compares earnings by gender. Finally, a rate is a more general measure in which one quantity is divided by another quantity, where both quantities don't necessarily need to have the same unit of measure. It can be equal to any value, including negative numbers. Like proportions, and it can be expressed as a number or a percentage. A well-known example is the annual population growth rate.

Knowledge check

(Table containing the population estimates, on July 1, of females and males in the canadian population between 2018-2020 in millions of people. Female : 18.7 (2018); 18.9 (2019); 19.1 (2020). Male : 18.4 (2018); 18.7 (2019); 18.9 (2020). Total : 37.1 (2018); 37.6 (2019); 38.0 (2020))

In the following slides, we will go through some example of proportions, ratios and rates from real data. Table 1 presents the demographic estimates in the middle of the year by sex for the years of 2018 and 2020. What proportion of the Canadian population was male in 2019? To find the answer, you would have to divide the number of males (18.7 million) by the total population (37.6 million). This gives us our answer: the proportion of males in the Canadian population in mid-2019 was 0.497. Multiplying by 100 will give the answer as a percentage 49.7 percent.

Now try to calculate the sex ratio, or the ratio of males to females, in the Canadian population in 2019. Pause the video and give it a try now. Looking at this table, we would say that the ratio is 18.7 million males for every 18.9 million females. However, this sounds confusing. There must be a better way to report this information! Instead, let's try expressing the ratio as a decimal number by dividing the number of males by the number of females. This gives us an answer of 0.99. In other words, in July 2019, there were 0.99 males for every female in the Canadian population. This would help your audience to understand that there were fewer males and females in the Canadian population. As well, given that this ratio is close to one, we can conclude that the gap between the number of males and females is small.

Finally, let's return to the same table to calculate the growth rate in the Canadian population between 2018 and 2019. First, you would calculate the difference in population between the two years, and then divide the difference by the population size in the earlier year, which in this case, is 2018. Multiplying this answer by 100 allows us to express this rate as a percentage. Here, we get a growth rate of 1.30%. Since the growth rate is positive, we can conclude that the Canadian population increased from 2018 to 2019.

Did you know? Rates can be used to predict the future!

(Table containing the population estimates, on July 1, of females and males in the canadian population between 2018-2020 in millions of people, as was shown in the previous slide. Female : 18.7 (2018); 18.9 (2019); 19.1 (2020). Male : 18.4 (2018); 18.7 (2019); 18.9 (2020). Total : 37.1 (2018); 37.6 (2019); 38.0 (2020))

Did you know that rates can also be used to predict the future? We can see in table 1 that the size of the Canadian population was 38 million in 2020, but if the data were not available yet, how would we be able to predict it from the 2018 and 2019 population estimates? A quick and simple way to do that would be to assume that the growth rate between 2019 and 2020 would be the same as the growth rate between 2018 and 2019. Then we would apply this rate to the 2019 population to calculate the growth rate, and add it to the 2019 population. As we saw earlier, the growth rate between 2018 and 2019 was 1.30%. Multiplying this rate by the 2019 population, 37.6 million, we get 0.5 million. This represents the predicted growth by mid-2020. Then, adding this number to the 2019 population, 37.6 million, we get 38.1 million. How did we do? The result, 38.1, is slightly larger than the real estimate of 38 million. This means that the actual growth rate between 2019 and 2020 ended up being smaller than it was in the previous year. Still, our prediction was very close.

Recap of key points

Let's recap the key points that we covered in this video. Proportions, ratios and rates are all useful for comparing numbers and can appear to be quite similar. However, a proportion is a part, share or number considered in comparative relation to a whole, while the ratio is the relative size of two values. A rate is the quantity that is divided by another quantity and all three measure have different properties. It is also interesting to note that rates ,such as the growth rates, can be useful in making predictions about the future.

(The Canada Wordmark appears.)

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