PART A
Pictograph
Definition of Pictograph
- Pictograph is a way of representing statistical data using symbolic figures to match the frequencies of different kinds of data.
Examples of Pictograph
- The pictograph shows the number of varieties of apples stored at a supermarket.
Solved Example on Pictograph
Use the pictograph to find the total number of apples stored in the supermarket.
Choices:
A. 150
B. 120
C. 140
D. 200
Correct Answer: A
Solution:
Step 1: The pictograph shows 14 full apples and 2 half apples.
Step 2: So, there are 140 + 10 = 150 apples stores in the supermarket.
Related Terms for Pictograph
- Data
- Symbols
| Pie charts are useful to compare different parts of a whole amount. They are often used to present financial information. E.g. A company's expenditure can be shown to be the sum of its parts including different expense categories such as salaries, borrowing interest, taxation and general running costs (i.e. rent, electricity, heating etc). A pie chart is a circular chart in which the circle is divided into sectors. Each sector visually represents an item in a data set to match the amount of the item as a percentage or fraction of the total data set.
|
Line Plot
Definition of Line Plot
- A line plot shows data on a number line with x or other marks to show frequency.
Examples of Line Plot
- The line plot below shows the test scores of 26 students.
The count of cross marks above each score represents the number of students who obtained the respective score.
Solved Example on Line Plot
Which of the line plots represents the prices of different story books given below?
$10, $30, $10, $10, $20, $30, $50, $40, $45, $50, $10, $30, $35, $30, $20, $40, $50
Choices:
A. Figure 1
B. Figure 2
Correct Answer: A
Solution:
Step 1: First arrange the data items from least to greatest.
$10, $10, $10, $10, $20, $20, $30, $30, $30, $30, $35, $40, $40, $45, $50, $50, $50
Step 2: Now group the data items that are the same.
4 story books cost $10
2 story books cost $20
4 story books cost $30
1 story book costs $35
2 story books cost $40
1 story book costs $45
3 story books cost $50
Step 3: Match the grouped data items with the figures shown. Figure-1 represents the data correctly.
Related Terms for Line Plot
- Number Line
- Data
- Frequency
Scatter Plot
Scatter plots show the relationship between two variables by displaying data points on a two-dimensional graph. The variable that might be considered an explanatory variable is plotted on the x axis, and the response variable is plotted on the y axis.
Scatter plots are especially useful when there is a large number of data points. They provide the following information about the relationship between two variables:
- Strength
- Shape - linear, curved, etc.
- Direction - positive or negative
- Presence of outliers
A correlation between the variables results in the clustering of data points along a line. The following is an example of a scatter plot suggestive of a positive linear relationship.
Example Scatterplot
Scatterplot Smoothing
Scatter plots may be "smoothed" by fitting a line to the data. This line attempts to show the non-random component of the association between the variables.
Smoothing may be accomplished using:
- A straight line
- A quadratic or polynomial line
- Smoothing splines - allow greater flexibility in nonlinear associations.
The curve is fitted in a way that provides the best fit, often defined as the fit that results in the minimum sum of the squared errors (least squares criterion).
The use of smoothing to separate the non-random from the random variations allows one to make predictions of the response based on the value of the explanatory variable.
Cause and Effect
When a scatter plot shows an association between two variables, there is not necessarily a cause and effect relationship. Both variables could be related to some third variable that explains their variation or there could be some other cause. Alternatively, an apparent association simply could be the result of chance.
Use of the Scatterplot
The scatter plot provides a graphical display of the relationship between two variables. It is useful in the early stages of analysis when exploring data before actually calculating a correlation coefficient or fitting a regression curve. For example, a scatter plot can help one to determine whether a linear regression model is appropriate.
Bar graph
Bar graphs are a very common type of graph best suited for a qualitative independent variable. Since there is no uniform distance between levels of a qualitative variable, the discrete nature of the individual bars are well suited for this type of independent variable. Though you can extract trends between bars (e.g., they are gradually getting longer or shorter), you cannot calculate a slope from the heights of the bars.
One Independent and One Dependent Variable
1. Simple Bar Graph
Here the Factory is our independent variable, since there is no unit of measurement for factories and no 'order' to the factories, the independent variable is nominal. The dependent variable is scalar, measured in defects/1,000 cars. Since the scalar dependent variable has a natural zero point (i.e., absolute or ratio), all of the bars are anchored to the horizontal axis, giving a common point of measurement.
2. Horizontal Bar Graph
Bar graphs can be shown with the dependent variable on the horizontal scale. This type of bar graph is typically referred to as a horizontal bar graph. Otherwise the layout is similar to the vertical bar graph. Note in the example above, that when you have well-defined zero point (ratio and absolute values) and both positive and negative values, you can place your vertical (independent variable) axis at the zero value of the dependent variable scale. The negative and positive bars are clearly differentiated from each other both in terms of the direction they point and their color.
3. Range Bar Graph
Range bar graphs represents the dependent variable as interval data. The bars rather than starting at a common zero point, begin at first dependent variable value for that particular bar. Just as with simple bar graphs, range bar graphs can be either horizontal or vertical. Notice in the horizontal example above, a reference line is used to indicate a common key dependent variable value.
Histogram
Histograms are similar to simple bar graphs except that each bar represents a range of independent variable values rather than just a single value. What makes this different from a regular bar graph is that each bar represents a summary of data rather than an independent value. For this type of graph, the dependent variable is almost always a scalar scale representing the count, or number, of how many of a sample fall within each range of the independent variable. In the example above, the sample is all the females in the U.S. The independent variable is age, which as been grouped into ranges of 5 years each. You should try and keep the ranges for each bar uniform (5 years in this case), with the exception possibly being the first and/or last range.
Two (or more) Independent and One Dependent Variable
1. Grouped bar graph
Here, we have taken the same graph seen above and added a second independent variable, year. The initial independent variable, factory, is nominal. The second independent variable, year, can be treated as being either as ordinal or scalar. This is often the case with larger units of time, such as weeks, months, and years. Since we have a second independent variable, some sort of coding is needed to indicate which level (year) each bar is. Though we could label each bar with text indicating the year, it is more efficient to use color. We will need a legend to explain the color coding scheme. Note that all of the bars for each level of factory are touching each other, indicating visually that they are grouped together.
2. Composite bar graph
Another alternative for a bar graph with two independent variables is to have the bars stacked rather than side-by-side. This arrangement is useful when the summation of all the levels of the second independent variable is as or more important than the values for each level. In the upper example, it is very easy to read the summed weight of all of the different materials in each sample. There are, however, tradeoffs. The stacking of the bars means there is no common baseline for the individual bar elements, making it hard to make direct comparisons for the subcategories. For example, it is hard to compare the iron content of the three samples. A particularly powerful use for the composite bar graph is when the sum of all the dependent variable values for each bar is the same, such as when the values are a fraction of a whole. In the bottom example, the sum of the three different types of fats will always equal 100 percent. With this layout it is easier to see the relative portions, if not the absolute values, of a particular fat type across oils.
Excel Tips
For information on creating bar graphs with Excel, go to the Bar Graphs Module, or go to the Excel Tutorial Main Menu for a complete list of modules.
Specific tips for bar graphs
- Vertical bar graphs are called column graphs in Excel
- Horizontal bar graphs are called bar graphs in Excel
- The clustered sub-type will do single bar and grouped bar graphs
- The stacked subtype will do composite bar graphs
- Range bars cannot easily be done in Excel without additional custom graph types loaded
- Double-clicking on the bars will allow you to set your preferences for bar display. Under the Options tab you can set the ratio of the bar width to gap between bars
The Histogram
The histogram is a summary graph showing a count of the data points falling in various ranges. The effect is a rough approximation of the frequency distribution of the data.
The groups of data are called classes, and in the context of a histogram they are known as bins, because one can think of them as containers that accumulate data and "fill up" at a rate equal to the frequency of that data class.
Consider the exam scores of a group of students. By defining data classes each spanning an interval of 10 points and counting the number of scores in each data class, a frequency table can be constructed as in the following example:
Frequency Table
| Group | Count |
| 0 - 9 | 1 |
| 10 - 19 | 2 |
| 20 - 29 | 3 |
| 30 - 39 | 4 |
| 40 - 49 | 5 |
| 50 - 59 | 4 |
| 60 - 69 | 3 |
| 70 - 79 | 2 |
| 80 - 89 | 2 |
| 90 - 99 | 1 |
To construct the histogram, groups are plotted on the x axis and their frequencies on the y axis. The following is a histogram of the data in the above frequency table.
Histogram
Information Conveyed by Histograms
Histograms are useful data summaries that convey the following information:
- The general shape of the frequency distribution (normal, chi-square, etc.)
- Symmetry of the distribution and whether it is skewed
- Modality - unimodal, bimodal, or multimodal
The histogram of the frequency distribution can be converted to a probability distribution by dividing the tally in each group by the total number of data points to give the relative frequency.
The shape of the distribution conveys important information such as the probability distribution of the data. In cases in which the distribution is known, a histogram that does not fit the distribution may provide clues about a process and measurement problem. For example, a histogram that shows a higher than normal frequency in bins near one end and then a sharp drop-off may indicate that the observer is "helping" the results by classifying extreme data in the less extreme group.
Bin Width
The shape of the histogram sometimes is particularly sensitive to the number of bins. If the bins are too wide, important information might get omitted. For example, the data may be bimodal but this characteristic may not be evident if the bins are too wide. On the other hand, if the bins are too narrow, what may appear to be meaningful information really may be due to random variations that show up because of the small number of data points in a bin. To determine whether the bin width is set to an appropriate size, different bin widths should be used and the results compared to determine the sensitivity of the histogram shape with respect to bin size. Bin widths typically are selected so that there are between 5 and 20 groups of data, but the appropriate number depends on the situation.
Histograms and Boxplots
The histogram provides a graphical summary of the shape of the data's distribution. It often is used in combination with other statistical summaries such as the boxplot, which conveys the median, quartiles, and range of the data.
| YEAR | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | |
| ACCIDENT INJURY TYPE | Fatal Accidents | 5,440 | 5,230 | 5,378 | 5,634 | 5,674 | 5,604 | 5,711 | 5,672 | 5,952 | 6,218 |
| Serious Injury | 8,067 | 6,942 | 6,696 | 7,163 | 7,444 | 7,600 | 7,375 | 7,384 | 7,020 | 6,978 | |
| Minor Injury | 28,778 | 30,684 | 30,259 | 31,357 | 33,413 | 25,928 | 15,596 | 13,979 | 12,893 | 12,072 | |
| TOTAL ACCIDENT INJURY | 42,285 | 42,856 | 42,333 | 44,154 | 46,531 | 39,132 | 28,682 | 27,035 | 25,865 | 25,268 | |
REAL LIFE DATA STATISTIC OF ACCIDENT INJURY
HISTOGRAM
This histogram shows the statistic of accident injury. In 2001, has lower injury that gets fatal than in 2009. However, for minor injury in 2009 has lower than 2001. Besides, accident that cause serious injury in 2001 has highest value than other year. In 2005, this histogram shows that total accident injury that happened in that year is the highest.
LINE GRAPH
This line graph shows the statistic of accident injury. In 2001, has lower injury that gets fatal than in 2009. However, for minor injury in 2009 has lower than 2001. Besides, accident that cause serious injury in 2001 has highest value than other year. In 2005, this line graph shows that total accident injury that happened in that year is the highest. From year 2005 to 2009 shows decreasing total accident injury that happened in these year.
PIE CHARTS
This pie charts shows the statistic of accident injury according the sectors. The sectors shows the percentage for replace the values of accident injury. For fatal injury, in year 2001 and 2002 shows same percentage that is 9% while in year 2000, 2003, 2004, 2005, 2006, 2007 also have same percentage that is 10%.
Besides, for serious injury, the percentage for year 2001, 2003, 2004, 2005, 2006, 2007, 2008, 2009 are 10%. The highest percentage for accident that caused serious injury is in year 2000 which has 11% while the lower percentage is in year 2002 that is 9%.
However for minor injury that happened in accident, in year 2009 has lowest percentage which has 3% while in year 2003 has highest percentage that shows 14%.
For total accident injury result, the highest percentage is 13% in year 2005 while the lowest percentage is in years 2000, 2008, 2009 that are 7%.
vii.
A line graph plots continuous data as points and then joins them with a line. Multiple data sets can be graphed together, but a key must be used. The advantages for using line graph are can compare multiple continuous data sets easily and interim data can be inferred from graph line
A pie chart displays data as a percentage of the whole. Each pie section should have a label and percentage. A total data number should be included. The advantages are visually appealing, shows percent of total for each category, display relative proportions of multiple classes of data, show areas proportional to the number of data points in each category, summarize a large data set in visual form, be visually simpler than other types of graphs, permit a visual check of the reasonableness or accuracy of calculations, require minimal additional verbal or written explanation, be easily understood due to widespread use in business and the media, an informative way to depict proportional statistical information and allow for easy comparisons.
A histogram displays continuous data in ordered columns. Categories are of continuous measure such as time, inches, temperature, etc. The advantages this histogram is visually strong, can compare to normal curve, usually vertical axis is a frequency count of items falling into each category, begin to show the central tendency and dispersion of a data set, closely resemble the bell curve if sufficient data and classes are used, show each interval in the frequency distribution, summarize a large data set in visual form, clarify trends better than do tables or arrays, estimate key values at a glance, permit a visual check of the accuracy and reasonableness of calculation, be easily understood due to widespread use in business and the media, and use bars whose areas reflect the proportion of data points in each class.
PART B
(a) Select a random sample of 25 IPMTAA’s students and get their shoe sizes, and construct histogram.
| Size of shoes | No.of students |
| 5 | 5 |
| 6 | 2 |
| 7 | 9 |
| 8 | 5 |
| 9 | 4 |
(b) Increase your sample size to 50 students and construct a new histogram.
| Size of shoes | No.of students |
| 5 | 8 |
| 6 | 6 |
| 7 | 17 |
| 8 | 11 |
| 9 | 8 |
(c) Increase your sample size to 100 students and construct a new histogram.
| Size of shoes | No.of students |
| 4 | 8 |
| 5 | 16 |
| 6 | 15 |
| 7 | 27 |
| 8 | 23 |
| 9 | 11 |
(d) Increase your sample size to 200 students and construct a new histogram.
| Size of shoes | No.of students |
| 4 | 9 |
| 5 | 28 |
| 6 | 34 |
| 7 | 49 |
| 8 | 53 |
| 9 | 21 |
| 10 | 6 |
What is the difference illustrated by the histogram in (a) and in (b).
Histogram in (a) shows the sample of 25 IPMTAA’s students of their shoe sizes while histogram in (b) shows the sample size of 50 students. For histogram (a), the sample of shoe for size 7 is the highest frequency that is 9 students and the lowest frequency is size 6 which is only 2 students. In histogram (b), the highest frequency still size 7 that is 17 students and the lowest is size 6 which is 6 students. The difference between these two graph because histogram in (b) increase the sample 25 more students from histogram in (a).
What does the histogram in (c) illustrates as the sample increase?
Histogram in (c) increase another sample of 50 students from histogram in (b). Because of the sample increase, the scale for maximum point become more greater. Besides, the size of shoes also have addition for another size which is size 4. The size of shoes for size 7 also experienced increase because of the sample become increase. For conclusion, the majority Malaysian’s shoes size is size 7.
In histogram (c), if you were told the graph down the middle, what will happen to the left and right side? Does the histogram is a symmetric graph?
In histogram (c), when the graph down the middle, the left and right side shows unbalanced graph. So that, the histogram is not a symmetric graph.
PART C
REFLECTION
I am Nur Lyiana bt Hamzah, a student of PPISMP BM/PJ/KS (2) unit. For this third semester, I was given a coursework on doing survey for Basic Mathematics subject by my subject lecturer, En.Nazri . I worked in a group of six in order to complete this coursework.
I had earn a lot of benefits from doing this Basic Mathematics Coursework. First of all, I searched from internet to find definition several of graph and from that I can recognize well to differentiate all of the graph. I do not have any difficulties to find abut the following graph which are pictograph, pie chart, line plots, scatter plots, bar graph and histogram because I from internet I can search multiples resources.
Besides that, this coursework practice me to explore and using Excel when construct the graph. For the beginning, I have a problem to construct the graph using the excel because I’m not always use this Excel in my task but my friends help to cover this problem. For task part B, me and my group co-operate together for doing survey the shoes’s size of IPGMTAA students. We chose unit PPISMP TESL third semester and PISMP SC second semester to complete our survey. We do not have any difficult to complete this survey because they give good cooperation.
After doing survey, all of group gather the data for easy we to construct the histogram. Once again, we must construct the graph using Excel. At this time, I just have a little problem to make sure all of the histogram have same scale. But, after I explored well that excel, finally I successed doing that graph according lecturer request. Then I have difficulties to answer several question in this coursework, so I ask my lecturer to have a guidance from him.
Moreover, this task also help me in future as I’m going to be teacher. I can teach my students all this methods and strategies. It is applicable in studying skills to give a better understanding to the students.
Lastly, I hope I will get this type of coursework in my next semester to improve my knowledge and experience on any problems and also to prepare myself to face the obstacles and challenges in my future.
Thank You.
BIBLIOGRAPHY
www.jkjr.gov.my/statistik.htm
excel-charts-the-chart-wizard-tutorial.html
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