what's going on besties today we're going to be tackling the t7 mathematics portion of the exam and we're going to be talking about tables charts and graphs let's get started the atits is going to test you on different graphical display types and how you use them there are five different types of displays that you're going to encounter when you take your te's one is the cartisian coordinate scatter plot line P Circle and bar graphs let's break each one of these down starting with cartisian coordinate chart also known as cartisian graph consists of two perpendicular lines or axes the horizontal axis is called the x axis and the vertical one is called the Y AIS their intersection at zero creates An Origin which is the starting point of plotting data as an example let's look at how we would plot 5 comma 4 so starting at our origin 00 0 right here in the middle if you look to the left and right of the origin Point you're going to see that you have negative numbers that move towards the left and you're going to have positive numbers that move towards the right it's very similar if you're looking at a number line in our example 54 the first data point is a positive number so we would move to the right five spaces on our x axis when you're looking at our Y axis we can see that the origin point 0 0 has kind of the same thing like we saw their number lines in our x axis so as we move up our number line it becomes more positive and as we move down our number line we become more negative our next data point is a positive number so we're going to move up four spaces to get to four now we're just going to place a DOT at this position of our plot and that is our 54 on our cartisian coordinate chart not too bad right next let's look at Scatter Plots so this type of graph is essential when we want to examine relationships between two different data sets or numerical variables essentially it's about observing how one variable correlates to the other let's consider the example of our scatter plot found here on the right side of your screen it investigates the amount of ice cream sales dependent on environmental temperatures as you can see over time the hotter the temperature outside becomes the more sales of ice cream increase Scatter Plots do hold significant value in scientific studies they are particularly useful when we need to alter one variable known as our independent variable and observe its impact on the other called our dependent variable when we're setting up our scatter plot it's crucial that we place that independent variable on our x axis this would be our X for instance if you're conducting an experiment where you vary the amount of fertilizer given to a plant to measure its growth over time the amount of fertilizer our independent variable would go on the x- axis the growth of our plant the dependent variable would be along our y AIS by doing it this way you can effectively analyze the correlation between these two factors as you can see the amount of fertilizer increases the growth of the plants now let's look at a line graph so a line graph is a ideal in observing changes over time so take for instance examining the produce sales over 9 months in this scenario the timeline is plotted along our X AIS showing us the individual months while the financial value values are marked on our y AIS showing the sales in the thousands in a scientific context we may be studying the respirations of a nursing student during clinical check off a line graph proves invaluable for this information if you're measuring how much oxygen the nursing student consumes over a period of time from just going into checkin during and then after a line graph will notice and illustrate these changes over time this makes it a perfect tool for visual izing and analyzing data that varies with time this is probably one of the most familiar charts and this is a pi or circle chart they use both names interchangeably as you can see from our pride chart that we have here as our example there was a survey that was completed in 2020 for nurses who were leaving a facility the data collectors collected information based on their reasons for departure and these were their results as you can see from our pie chart each color corresponds to a different reason so for example people leaving for personal reasons was about 50% those who transitioned to a competitor was about 20% travel nursing was about 133% no opportunity to advance was 10% and then lastly those individuals who did not provide a reason was 7% a bar graph is particularly useful when we need to compare multiple groups or categories for example in our first set of bar graphs you can see that it illustrates the number of worldwide electric cars per year as you can see as the years progress the number of electric cars also increase going back to our plant experiment that we talked about before imagine that we're in a laboratory setting investigating how various light colors affect the rate of photosynthesis in such a case a bar graph would be an excellent choice for this experiment we could plot the rate of photosynthesis here on our y AIS with each color represented as a distinct bar down here on our X AIS often in bar bar grass each bar signifies the average or mean derived from the collected data this makes barrass a useful tool and a very powerful tool at that for visualizing representations of comparing average values across different categories or groups here is a quick and useful tool as to why we would use different graphical displays and what specific cases we would use them as it's a little cheat sheet to help you pass your aits when you come across these questions on the exam all right once we've determined the graph to use we can begin the graphing process let's experiment with this concept using hypothetical data from an ice cream sales that we discussed earlier imagine that we have data showing the average temperature in an area and the corresponding ice cream cone sales during different months if we want to illustrate the total ice cream sales in each month as a proportion of the year's total sales a part chart would be ideal each slice of the pie chart would represent a different month showing us its average s sales that they share over that year on the other hand if our focus is tracking the average high temperatures in an area over time a line graph would be the way to go this graph would effectively depict temperature Trends throughout the year now let's say that we're interested in comparing ice cream sales across specific months in this scenario a bar graph would be highly effective it would allow us to compare the ice cream sales to the specific months side by side providing a clear visualization of the comparison let's dive into the Practical creation and interpretation of graphical displays this is very important for you to know for the te's and we're going to start by focusing on Scatter Plots using our previous data set here is how we would construct the scatter plot and what each element represents the very first thing that we want to do is we want to establish our axes so on the xaxis we're going to plot the average high temperatures and on the Y AIS we're going to plot our ice cream sales it's important that you make sure that you label each axes clearly including the unit of measures that you're using for instance the x axis might be labeled average high temperature and we're using the unit of measure of Fahrenheit and on the Y AIS it is our ice cream sales and we're doing cones per month we also want to make sure that we title our graph a good title should encapsulate the essence of the graphs content it's not enough to just have a vague title like ice cream right instead it should be informative such as correlation of ice cream sales and average high temperatures 2023 this title should essentially narrate the story of the data that we're trying to present after labeling our chart we can now plot the data points in a scatter plot each point represents a data pair correlating between temperature with sales the arrangement of these points can reveal patterns or correlations an important feature of our scatter plot is our best fit line this line represents the average Trend through our data points when drawing at freehand we aim to position it so that roughly half the points lie above it and the other half lie below it remember this line should not go beyond our range of data as it represents interpolation within the existing data set avoid putting any kind of arrows or extending them past the outermost data points in addition in terms of SC scaling we want to ensure that the increments of each axis is uniform and clear for instance if each major grid line of the xais represents an increment increase of 15° C this should be consistently applied throughout our line in our case with this graph the ice cream sales in average temperature increase in increments of 10 lastly remember that including zero on your graph is not always necessary if your data doesn't include or approach zero it does not need to be present on your graph this helps keep the focus on the relevant data ranges and avoids misleading representations one of the things that the teas loves to test you on is bad graphs and misrepresentations what are we missing what could be better about our graphs let's take a look at the example of a bad graph focusing on one that's supposed to show how different amounts of fertilizers affect plant growth when you look at this you immediately notice several errors firstly the title of the graph while is present isn't descriptive enough it needs to clearly indicate what is being plotted on our X and Y axes such as the relationship of fertilizer to plant growth next there's a major issue with how the axes are set up plant height is actually listed down here on our x axis but since we have varying amounts of fertilizer that actually should be our independent variable and should be placed down here on our x axis and that plant height should be moved to our y y AIS because it is our dependent variable it's also crucial to include units for each variable but as you can see they're missing here plant height should be labeled on the Y AIS with appropriate units and fertilizer amounts should also be labeled on the xaxis with its amounts of units the scaling of these axes is another problem the x-axis shows non linear scaling numbers of 5 12 15 19 23 and 30 and a well-made graph the distance between each point should be consistent uniform and clear additionally only having a single number on our y AIS 13 doesn't help with accurate measurements either lastly our best fit line on this scatter plot extends past our range of data this is also going to be incorrect that best fit line should only Encompass the range of data points on our graph in summary the graph has quite a few significant errors demonstrating what to avoid when creating or interpreting a scatter plot let's take a look at our first practice question which of the following descriptions best matches a scatter plot is it a a graph that uses vertical or horizontal bars to show comparisons among categories b a graph that displays data points along a line typically used to show changes over time c a circular graph divided into slices to illustrate numerical proportions or is it d a graph that displays data points to represent the relationship between two variables and the the correct answer is d a graph that displays data points to represent the relationship between two variables as we know a scatter plot is characterized by its use of data points plotted on a cartisian coordinate system showing how one variable is affected by the other which is why D is the most correct answer you were tasked with presenting the monthly sales data for three different products over the course of a year which type of graph would be the most effective in displaying this information is it a a High chart b a line graph c a scatter plot or d a cartisian coordinate and the correct answer is b a line graph as we know a line graph is ideal for showing changes over time making it the best choice for presenting sales data across different months it allows the audience to easily see Trends and compare the performance of the three products throughout the year pie charts are more suited for showing parts of a hole at a single point in time scatter plots are used to show relationships between two variables and cartisian coordinates would not be used in the scenario a bar graph is used to compare the average test scores of four different classes algebra biology chemistry and English sounds like the TEAS test right however the bars are not labeled with the names of the classes how does this affect the interpretation of the graph is it a it allows for a straightforward comparison of the test scores across subjects B it does not impact the interpretation as long as the scores are visible C it creates ambiguity making it difficult to attribute the correct scores to each class or is it D it improves the visual Simplicity of the graph focusing attention solely on the numerical scores and the correct answer is C it creates ambiguity making it difficult to attribute the scores for each class without these labels indicating which bar represents which classes it becomes impossible to determine the average test scores for each individual class of algebra biology chemistry and English now let's talk about linear exponential and quadratic graphical Trends in order to understand these three different kinds of Trends we're going to put our imagination hats on so let's say that we're a scientist who are observing Turtle populations in a series of islands focusing first on two islands funky Island and get down Island in what we term Turtle Euro Z three turtles each make their way to Funky and get down Island upon visiting each island after 1 year we note that funky Islands Turtle count has risen to five while get down islands has reached six the following years F's population which is our funky island has grown to seven and our G island which is get down island has impressively doubled to 12 year after year we maticulously record our findings aiming to discern any emerging patterns we Ponder whether the Turtle populations expand at a steady Pace a concept known in mathematics as constant difference let's take a closer look at each one of these different kinds of islands and graph Trends so starting with funky Island we started with three turtles if two more arrived each subsequent year that after a decade the calculation for Island F would be the initial three Turtles plus two turtles for each of the 10 years totaling 23 as we generalize this if x equals the total number of years elapsed the turtle population on island F would be represented as 2x + 3 which intriguingly forms a linear equation y = mx + b where m is the constant rate of increase and B is the starting number of the turtles upon graphing this relationship of funky Island we discover a linear Trend affirming our hypothesis this linear model is characterized by a constant increase or decrease turning our attention to the turtles on get funky Island G it's another intriguing pattern that emerges their numbers seem to double each year indicating exponential growth rather than linear growth unlike that steady addition that seemed to take place on funky Island get down Islands Turtles increased by a factor of two annually to illustrate the population for the seventh year we would be calculating that by taking the initial three Turtles and D leveling them seven times a process conveniently expressed using an exponent this kind of a pattern aligns with exponential function which is y = a * B with the exponent of X where it represents the starting numbers of turtles b as the growth rate in this case two and X is the number of years our analysis is confirmed as depicted in the graph of get down Island's Turtle population as you can see we have a nice Cur steadily increasing over time our findings highlight a fundamental principle of growth patterns given enough time any exponential function with a growth multiplier greater than one will surpass a linear function lastly let's focus our efforts on quickstep island initially there was only one Turtle but by year one the count reached 13 by the second year the number stored to 45 and by the third year it stood at 97 these patterns present a curious pattern that seems to correlate with the passing of years the turtle population on quickstep island can be described by the formula y = 10 x^2 + 2x + 1 where X represents the number of years this means that for the 9th year the population would be 10 * 9 years sared + 9 * 2 + 1 doing our calculations we have 92 is = to 81 1 10 * 9 is = to 810 2 * 9 is = 18 and then of course we bring up the number one so if we add all of these numbers together we're going to get a total of 829 turtles by the time that we reach our 9th year this pattern is indicative of a quadratic function which takes the form of Y is = a * x^2 + BX + C in this case b is equal to 2 and C is equal to 1 the graph of this function reveals that classic U-shaped curve that's also symmetrical and distinctive when we're looking at quadratic equations when complimenting which Turtle population would outgrow the others over time it might be tempting to say that the quadratic growth of quickstep Island would be the given Choice given its rapid Ascent however it's important to remember that exponential growth when it comes to capacity it is going to increase indefinitely it will eventually surpass that quadratic growth the quadratics function's growth is Tethered to the square of X whereas the exponential growth X serves as the exponent itself offering balance potential for increase let's take a look at how this will be displayed on the atits starting with our first practice question Calvin is selling apples for $2 each at a farmers's market he has written down his sales for the last 30 minutes as follows and as you can see over here on the right hand side of our screen this is our data points what type of relationship is depicted between the numbers of the apples bought and the total cost is it a linear B exponential C quadratic or D cubic and the correct answer is a linear the cost increases by a constant amount of $2 with each additional Apple indicating a constant rate change this consist consistant increment is characteristic of a linear equation where the graph is going to be a straight line Julie is growing bacteria collected by a bacterial sputum sample for a patient admitted with pneumonia she documents her findings every hour what type of graph best represents the relationship between the time and the number of bacteria is it a linear B exponential C quadratic or D cubic and the correct answer is B exponential the number of bacteria doubles with each passing hour indicating the bacteria count is not increasing by a fixed amount but rather a fixed proportion this pattern of growth where the rate of increase becomes progressively more rapid is indicative of an exponential relationship next let's take a closer look at directions of Trends in graphs when we're analyzing graphs it's important to understand that different Trends can depict either increasing decreasing or no change trends when we're looking at graph Trends we want to look from left to right where does the line start in the left and what is the trend when we move across to the right let's start with increasing Trends so graphs with increasing Trends show values that rise As you move from left to right as you can see from our two different examples if you look at the starting point on the left you can see that each line increases in value as we start moving to the right so as you can see here we start moving up in value and again on this example we also move up in our value so we're going to get a concave up or concave down increasing kind of distribution with decreasing trending graphs they depict values that fall as they progress from left to right on the graph so as you can see from our example as you look at the starting point on the left you're going to see that each line is going to decrease in value as we move to the right so up here we have a concave up decreasing graph meaning that we start high and then we come down low and then over here on the right we have a concave down decreasing graph again starting high and as we move to the right we lower the number lastly we have a no change Trends kind of graph so these graphs will show a flatline representation of a situation where there's no change over time the value remains constant regardless of the changes that are happening along the horizontal axis as you can see here we just have this flat line on our graph which shows that as we move left to right there is no change making it a no change graph outliers on graphs are data points that stand apart from the crowd catching your eye because they don't quite fit in with the rest imagine that you're looking at a graph and while most of the data points are in a cozy cluster like we see in the graph here on our screen there's one point or maybe even two points that are going to kind of miss the memo with the rest of our data points they're going to be sitting off far to the side above or below our best fit line as you can see here in our example sometimes these outliers are a result of a simple slip up maybe a typo or data that was entered incorrectly or even a glitch in the measurement equipment that we're using other times these are real deal genuine deviations that tell a story of their own like a sudden spike in sales after a viral marketing campaign or an unexpected drop in temperatures on an otherwise warm month if any of you live in Florida you know exactly what I mean these types of questions on the t's tend to be easier because we can visually see the data points and can easily denote that something is wrong we introduced dependent and independent variables a little bit earlier but now we're going to dive a little bit deeper into understanding what they mean the independent variable is often seen as the cause is the variable that we manipulate or change to observe how it affects another quantity on the other hand the dependent variable is considered the effect it's the value that depends on on or is determined by the changes of our independent variable for instance in an experiment measuring the growth of plants based on the varying amount of water the amount of water given that is going to be our independent variable is going to directly influence the growth our dependent variable so let's start with the basics when it comes to graphing these variables the independent variable which is often ploted on our X AIS as we know is the variable that we change in order to control an experiment think of it as the input of our cause and effect relationship on the flip side we have our dependent variable which is represented here on our y AIS and this is the outcome or the effect that we are going to observe as a result of the changes to our independent variable so essentially the value of the dependent variable is going to hinge on the influence of our independent variable if you're trying to figure out how independent and dependent variables affect an equation it'll looks something similar to this we have the equation y = 3x + 5 based on our equation we know that Y is going to equal the outcome of the value because it is the lone variable on the other side of our equal sign that means that X is going to be our independent variable we have control over what we're going to put in X which is ultimately going to influence r y let's take a look at another example imagine a hospital system is conducting an experiment to understand the relationship between nurse to Patient ratios and average recovery time for patients on an orthopedic unit the task is to represent the given data on a graph this scenario poses an interesting question between the nurse to Patient ratio and the average recovery time which is our dependent variable and which is our independent variable let's recall that our dependent variable is ultimately going to be influenced by our independent variable so when our case does the nurse to Patient ratio depend on the recovery time or is it the other way around given that in this experiment we can choose the nurse to Patient ratio and implies that the ratio is the variable within our control hence nurse to Patient ratio is going to be our independent variable which we are going to plot down here on our X AIS on the other hand the average recovery time which we aim to measure naturally is going to become our dependent variable which we're going to plot over here on our y AIS as you can see down here on our graph the higher the nurse the patient ratio is is ultimately going to increase the average recovery time for our patients lastly to test our knowledge from before what kind of relationship do we see when we're looking at these two variables is it linear exponential or quadratic that's right you've guessed it it's linear because we have a straight line good job in a study examining the effects of study hours on test scores researchers track the number of hours spent studying X and the results testing scores why given the relationship described in the equation of y = 2x + 70 where y represents the test scores and X represents study hours determine which variable is independent and which is dependent is it a x is dependent variable because it relies on the test scores B why is the independent variable because it is determined by the number of study hours is it c x as the independent variable because it represents the controlled amount of study time influencing the test scores or is it D why is a dependent variable because it dictates the number of study hours needed to achieve a certain score and our correct answer is C in this context the number of study hours which is X is the variable that the researchers control or manipulate to observe the effect on another variable making it our inde dependent variable the test scores which would be our y on the other hand are affected by the changes in our study hours making them the dependent variable the equation y = 2x + 70 clearly shows that Y which is our test scores is going to change in response to X which is our study hours underscoring the dependency of test scores and the amount of time spent studying and a clinical trial to assess the effectiveness of a new dietary supplement on improving blood pressure levels participants Baseline blood pressure readings are taken before starting a supplemental regimen and their daily supplemental dosage varies among participants based on the information provided which variable is independent and which is dependent is it a medication dosing is the dependent variable because it is influenced by the changes in blood pressure readings is it B blood pressure readings is the independent variable because they determine the dosage of dietary supplement is it C medication dosing is the dependent variable because it represents the dosage of the supplement which is manipulated to observe its effect on blood pressure or is it D blood pressure readings are the dependent variable because it depends on the dosage of the dietary supplement and the correct answer is D in this clinical trial the variable the researchers actively manipulate is the dosage of that dietary supplement given to the participants making that medication dosing our independent variable the purpose of varying that dosage is to observe its impact on a specific outcome which is in this case is going to be our blood pressure readings making that our dependent variable next let's talk about correlation and covariance so correlation and covariance is a statistical measure that describes the relationship between two variables like in all of our examples for mathematics we call these variables X and Y to illustrate at this let's imagine that we plot points on a graph using X and Y AIS and the pattern emerges somewhat like we see in our examples the way that it's depicted in our first example here suggests that positive correlation between X and Y that is that when X goes up Y is also going to Trend up as well now let's also consider a scenario where the inverse relationship is true like we see in our second example with negative correlation as X increases y is actually going to decrease causing an inverse relationship in our last example we have no correlation meaning that the points are plotted on the X and Y axes but they don't follow the same Trend or have any correlation as you can see from our graph we're unable to draw any kind of line through any of these plotted points to show a relationship among them this shows that there is no relationship between these points as they're scattered all throughout our graph a study was conducted to a explore the relationship between a number of hours spent studying per week and the resulting GPA of college students the data collected from a sample of students is presented in our table that we have right here on the right hand side of our screen based on the data provided what kind of correlation exists between the number of hours studied per week and the GPA is it a positive correlation a b a negative correlation C no correlation or D cannot be determined from the data provided and and the correct answer is a we have a positive correlation the table shows that the number of hours studied per week which is our X increases and the GPA which is our y also increases this upward Trend suggests that we have a positive correlation indicating that students who spend more time studying tend to have higher GPA so this next question can tend to be a little bit tricky so hang tight with me a fitness coach is analyzing the relationship between the number of calories consumed daily that's our X and the weight change Y in kilograms observed over a month in clients the following data is recorded in a table what does the data suggests about the co-variance between daily caloric intake and weight change do we have a positive co-variance a a negative covariance B C zero covariance or D the data is insufficient to determine covariance and the correct answer is B we have a negative co-variance so the table illustrates that as the daily calorie intake X increases the number change y shift decreases this pattern implies that the negative covariance between caloric intake and weight change indicates that a lower calorie intake is associated with less weight gain and more weight loss lastly we're going to highlight the concept of proportionality which fundamentally comes in two flavors direct and inverse proportionality let's start with direct proportionality in this scenario we use variables Y and X like we have always so far to describe a situation where y increases or decreases in tandem with X mathematically we express this relationship as Y is equal to KX where K represents proportionality constant this constant of K is crucial as it determines the specific relationship between y and x and uring that any change in y is going to be directly linked to the proportional change in X so what does this mean in real world scenarios so if you consider the relationship between the Distance by a car and the time it takes to travel that distance is a constant speed as the time increases the distance traveled also is going to increase if a car is traveling at a steady rate of 60 mes per hour the distance covered is directly proportional to the time traveled now let's pin a little bit to talk about inverse proportionality in contrast to direct proportionality when we say that Y is inversely proportional to X we're describing a situation where as y increases X is going to decrease and vice versa the mathematical representation for indirect or inverse proportionality relationships is y is equal to KX where K remains our proportionality constant however in this Formula K is position position above X and a fraction signifying the inverse relationship the setup implies that X as it becomes larger Y is going to get smaller and the opposite of true when X decreases a real ball example of this is a relationship between the amount of time taken to complete a task and the amount of workers assigned to that task if more workers are assigned to that task the time taken to complete the task is going to decrease and if fewer workers are assigned to that task then the time taken is going to increase this is a great example of inverse variation where the increase in the number of workers tends to decrease the amount of time that it takes to complete a task graphs representing direct and inverse relationships exhibit distinct characteristics based on the nature of the relationship between the two variables as we discussed in a direct relationship as one variable increases the other variable is going to increase at a constant rate this is reflect in our graph as a straight line if you plot y our dependent variable against X our independent variable for a direct relationship Y is equal to K * X the line will pass through the origin 0 0 the slope of this line is determined by that constant K and the line will Ascend to the right if K is positive indicating that y increases in X is also going to increase in an inverse relationship as one variable decreases the other a variable is going to increase this type of relationship is captured by our equation like we talked about before is y is equal to K overx and the graph is going to look significantly different from a direct relationship graph instead of seeing that straight line that we've become accustomed to the graph is actually going to feature a curve line that approaches the axes but it's actually never going to touch them as X increases the Y value is going to decrease approaching zero but never actually reach ing it reflecting that inverse relationship between the two variables a Biology experiment investigates the growth of a plant species under different light intensities the table below shows the height of the plants why after 30 days of various light intensities X based on the data provided which statement best describes the relationship between light intensity and plant height is it a as light intensity increases plant height decreases indicating an inverse relationship is it B there is no clear relationship between light intensity and plant height is it c as light intensity increases plant height increases at a constant rate suggesting a direct relationship or is it D plant height changes unpredictably with changes of light intensity and the correct answer is C the table shows a consistent pattern where we double the light intensity is going to result in doubling that plant's height this proportional increase increase indicates a direct relationship where the plant height is directly proportional to our light intensity a physics study examines the cooling time of a hot beverage as a function of the amount of cream that was added the table shows the cooling time Y for the beverage to reach a drinkable temperature for various amounts of cream added x what does the data suggest the relationship between the amount of cream added and the cooling time of the beverage is it a adding more cream significantly increases the cooling time showing a direct relationship is it B the cooling time decreases as more cream is added illustrating an inverse relationship is it C there's no significant change in cooling time with varying amounts of cream indicating no relationship or is it D the relationship between the amount of cream added and the cooling time is inconsistent and the correct answer is B the data indicates that as the volume of cream increases the cooling time of the beverage to reach drinkable temperature decreases this temperature suggests that the cooling time is inversely related to the amount of cream that's added as an increase in one leads to a decrease in the other I hope that this video is helpful in understanding everything that you're going to need to know when it comes to tables charts and graphs as always if you have any questions make sure that you leave them down below I love answering your questions head over to nurse chunk store.com there's a ton of additional resources to help you Ace that ait's exam and and as always I'll catch you in the next video bye