In science, it's not enough to take only one measurement and assume that it's correct. Whenever we measure something there are lots of things that can happen to make the measurement inaccurate. We try to overcome this problem by taking as many measurements as possible, then determining the average of all of the measurements. The more measurements we take the closer we get to the best possible answer.
Let's say for example that we measured the temperature outside on five different days. The temperatures were 25°C, 26°C, 24°C, 22°C, and 29°C. To calculate an average you start by adding together the values for all of the measurements you took:
25 + 26 + 24 + 22 + 29 = 126
You then divide this number by the number of measurements you took. In this case, we took five measurements so we divide 126 by 5 (NOTE: I almost always use the / symbol to mean divided by, and almost never ÷ but they mean the same thing).
125 / 5 = 25
This means that the average temperature for the five days that we took measurements was 25°C.
If you are using spreadsheet software such as Excel, Numbers or Sheets to do your work, you can take a shortcut when calculating averages. All you need to do is to type in the following formula:
Then, click inside the parentheses and highlight the cells for which you would like to calculate the average and hit enter. This can be a big time saver if you have a lot of averages to calculate! Below is a link to my YouTube tutorial on how to use Google Sheets to perform basic calculations including averages.
SKILLS CHECK Imagine that we wanted to find out the average age of the players on a football team. Their individual ages are 21, 22, 27, 31, 26, 23, 23, 25, 34, 20 and 30.
1. Calculate the average of the 11 players using a calculator. 2. Perform the same calculation using a the 'Average' function on a spreadsheet program such as Google Sheets.
The answer you should receive is 25.6 (rounded to one decimal place).
The Problem with Averaging
Averaging is a great way to get an accurate result if there is only one answer to a question, but it can cause problems when used in the wrong circumstances. An example of when averaging can go wrong is when the American Air-force was deciding on the design of the cockpit for some planes they were building during World War II. They wanted the equipment to be arranged correctly for the highest possible number pilots, so they measured the height of hundreds of pilots and built the planes based on the average height. The only problem was that none of the pilots were actually the average height, so the cockpit wasn't correctly designed for anyone! The Air-force learned that they needed to make seats adjustable so that all pilots could fit in them comfortably.
ACTIVITY Try taking a survey of the students in your class. Record their height, age in months, and the number of people in their family. Is anyone in your class the 'average' student?
We make graphs in science to help us identify relationships between our independent and dependent variables. In the Interpreting Graphs (Advanced) tab above you can find out how to use the equation for graphs as part of your interpretation. This section is just a basic introduction to some of the important things that you should be looking at. It is important to note that the following can not be applied when discussing nominal data (that is, data that uses words instead of numbers, such as comparing Coke to Pepsi). This information only applies to interval and ratio data (that is, data that falls on the same scale such as temperature or mass). Click here for more information on the difference between the different types of data. The Direction of the Graph The first thing to take note of is the direction of the graph. If it is going up (that is, as the independent variable increases so does the dependent variable) then we say the relationship is positive. If it is going down, we say that it is negative.
Examples of Positive Graphs
Example of a Negative Graph
The Shape of the Graph There are many different shapes that a graph can form, but at this stage we are going to limit ourselves to simply saying 'linear', 'curving up' or 'curving down'.
Example of Curving Up
Example of Curving Down
If you would like some information on how to work with graphs in Google Sheets you can follow the links below to my tutorial videos on YouTube:
In the Interpreting Graphs (Basic) section I wrote that we use graphs in order to identify relationships between independent and dependent variables. Ideally, we will be able to do this using an equation that we derive from the graphed data. This equation can then be used to carry out calculations. Below are two links to tutorial videos on how to use Google Sheets to graph data and determine an equation for the trendline:
Linear Proportional It is often easy to see a linear graph and think that the y and x values are proportional, however, a graph is only proportional if the y intercept (c value) is equal to zero. So the general equation for a linear proportional relationship is y = mx. In a linear proportional graph, m represents the value of the gradient.
Linear As the name suggests, linear graphs form a straight line. As opposed to proportional graphs, linear graphs may have a y intercept that is greater than or less than 0, so the general equation for linear graphs is y = mx + c, with c representing the y intercept.
Negative Linear Resist the urge to call this a negative proportional graph! As mentioned above, graphs are only proportional if the y intercept value is 0. The general equation for a negative linear graph is y = -mx + c.
Power A power graph (not to be confused with an exponential graph) increases as a curve because the x value is being raised to a power. The general equation for a power graph is y = kx^n. Here, because of formatting issues I've had to use the ^ symbol to mean 'raised to the power of'. You might notice that I've used k here instead of m, this is because k does not represent the gradient in this case.
Inverse Inverse graphs have the general equation of y = n/x. However, when you see this relationship represented as a trendline equation on Sheets or Excel it will be in the format y = nx^-1. This is because nx^-1 = n x 1/x = n/x. It is possible to have a relationship that is inverse proportional, but as was the case with linear graphs, a relationship is only proportional if there is no c value (constant).
Exponential As I mentioned above, it can be very easy to confuse a power relationship with an exponential relationship because they are both curved, but there is a very important difference between the two. While the equation for a power graph is y = x^n, the formula for an exponential graph is y = n^x. It is uncommon in high school science to come across an exponential relationship, so if your data is generating a curved graph, you are safer to assume that it is a power relationship.
Log The general equation for a log relationship is y = logn(x). It is rare that you will encounter a log relationship unless you are investigating certain areas such as pH.
SKILL CHECK Use the shape of the graph below as well as the equation shown in the top right hand corner to determine the type of graph.
Sometimes it can be a little intimidating when you look at a worded problem in math or science and have no idea how to answer it. Examiners like to pose questions in unfamiliar ways because it shows that you really understand the problem and are not just going through the motions, but we can still take a systematic approach to answering this type of question. Here is my step by step approach to solving problems involving math:
List the known information.
List information that you can assume or infer.
Put a question mark next to information that you need to know.
Find an equation for which you have all necessary pieces of information except one.
Use the equation to find the unknown information.
Let's use this question about Ohm's Law as an example.
A student is conducting an investigation into the relationship between current, resistance and voltage. They set up a simple circuit with a 1.5 volt battery and a 10 Ohm resistor. Calculate the current flowing through the circuit.
Here are three equations that you might learn in a unit on electricity and hence might appear on a formula sheet.
Charge = Current x Time Power = Volts x Current Volts = Current x Resistance
1. We start by listing the information that we know because it was stated in the question:
Volts = 1.5 V Resistance = 10 Ohm
2. Since this is a fairly simple example, I haven't included a need to infer any information. But I'll give another example below that requires a little reading between the lines.
3. Next, write down a question mark next to the information that we need to know, in this case the current.
Volts = 1.5 V Resistance = 10 Ohm Current = ?
4. Looking at our choices of equations you can see that the first two are not suitable because they both contain two unknowns (either charge and current or power and current), but in the third equation the only information that we are missing is the value for current, so this is the equation that we need.
5. Volts = Current x Resistance 1.5 V = Current x 10 Ohm Current = 1.5 V / 10 Ohm = 0.15 A
Reading Between the Lines As mentioned above, sometimes information can be hidden in the wording of a question, so it's important to read each question carefully. An example of an inference that you could make when reading a question is below.
A car is traveling at 50 km per hour when it suddenly slams on the breaks and comes to a complete stop.
Even though it's not written in the question, we can infer that the speed of the car after it stops is 0 km per hour, because a stopped car has no speed. So even though it looked like there was only one number given in this question, there were actually two.