Probability and Statistics Chapter 1.1 Review of of Descriptive Statistics

Main Ideas

The video introduces several ways to organize and display data: categorical and grouped frequency distributions, histograms, frequency polygons, ogives, and pie charts. These methods make data easier to interpret and compare.

1. Categorical Frequency Distribution

A categorical frequency distribution is a table listing categories and how often each occurs. The example groups 35 cadets by blood type.

How to construct one:

1. List each category in a column.
2. Go through the raw data and tally each value under its category. Mark each value as it is counted to avoid counting it twice.
3. Count the tallies to find each category’s frequency.
4. Find the relative frequency:

[ \text{Relative frequency}=\frac{f}{N}\times100\% ]

where (f) is the category’s frequency and (N) is the total number of observations.

1. Check that the frequencies sum to (N) and that the relative frequencies total approximately 100%. Small differences may result from rounding.

In the blood-type example, the frequencies are A: 8, O: 14, B: 8, and AB: 5.

2. Grouped Frequency Distribution

A grouped frequency distribution organizes numerical data into class intervals and records how many values fall in each class.

How to construct one:

1. Form the class intervals.

   • Find the highest and lowest values.
   • Compute the range:

   [ R=\text{highest value}-\text{lowest value} ]

   • Choose the number of classes. The video suggests that researchers commonly use between 5 and 12 classes, although a problem may specify the number.
   • Find the class width:

   [ i=\frac{R}{\text{number of classes}} ]

   Round the result up to a whole number.

   • Set the first class’s lower limit at the lowest data value, or at a slightly lower value if appropriate. Calculate its upper limit as:

   [ \text{upper limit}=\text{lower limit}+\text{class width}-1 ]

   Continue the intervals consecutively, with no gaps or overlaps.

2. Tally each data value in its appropriate class.

3. Count the tallies to find each class frequency.
4. Calculate each class’s relative frequency using (f/N \times 100\%).
5. Check that the class frequencies sum to the number of observations and that the relative frequencies total approximately 100%.

In the GRE-score example, the range is (780-746=34). For five classes, (34/5=6.8), which rounds up to a class width of 7.

3. Histograms

A histogram represents a frequency distribution with adjoining rectangles.

• The horizontal axis shows the data values or class intervals.
• The vertical axis shows frequency, relative frequency, percent frequency, or probability.
• The rectangles touch because the classes are continuous.
• A histogram can help reveal the distribution’s shape, center, and spread.

For intervals with shared limits, each rectangle spans its class interval, with a height equal to the class frequency. If the class limits have gaps—for example, 90–98 followed by 99–107—find the class boundaries so the bars join continuously.

Finding class boundaries:

• The boundary between two consecutive classes is halfway between the previous class’s upper limit and the next class’s lower limit.
• For whole-number limits such as 98 and 99, the boundary is 98.5.
• Find the first lower boundary and final upper boundary by extending the same half-unit adjustment. For example, the interval 90–98 becomes 89.5–98.5.

Draw each rectangle from its lower to upper class boundary, using frequency as its height.

4. Frequency Polygons

A frequency polygon plots class frequencies at the midpoints of their intervals and connects the points with straight lines.

How to construct one:

1. Find each class midpoint:

[ \text{midpoint}=\frac{\text{lower limit}+\text{upper limit}}{2} ]

1. Plot each midpoint on the horizontal axis and its frequency on the vertical axis.
2. Connect the points with straight lines.
3. Add a point with frequency zero before the first class and after the last class, then connect these points to the ends of the graph to close the polygon.

5. Ogives (Cumulative Frequency Graphs)

An ogive plots cumulative frequencies against class boundaries. It shows how many observations are at or below successive values.

How to construct one:

1. Calculate cumulative frequencies by adding the class frequencies successively.
2. Plot each cumulative frequency against the corresponding upper class boundary.
3. Add a starting point at the lower boundary of the first class with cumulative frequency zero.
4. Connect the points with straight lines.

In the example, frequencies of 3, 10, 15, 7, and 5 produce cumulative frequencies of 3, 13, 28, 35, and 40.

6. Circle Graphs (Pie Charts)

A circle graph, or pie chart, shows how a whole is divided into categories. Each sector’s size is proportional to its category’s share of the total.

How to construct a pie chart from frequencies:

1. Add the frequencies to find the total (N).
2. Calculate each category’s percentage:

[ \text{Percentage}=\frac{f}{N}\times100\% ]

1. Calculate each sector’s central angle:

[ \text{Angle}=\frac{f}{N}\times360^\circ ]

1. Draw the sectors using the calculated angles, and label them with their categories and percentages.
2. Check that the percentages total approximately 100% and the angles total (360^\circ).

The video illustrates that a sector representing 8 hours of a 24-hour day takes up (8/24=1/3) of the circle. Another example converts percentages to fractions and angles: 50% corresponds to (180^\circ), while 25% corresponds to (90^\circ).

Speakers and Sources

• One unnamed instructor or presenter provides the explanations and worked examples throughout.
• Music is heard in the background. No additional speakers or cited sources are identified.