Sections 8-9 Histograms, Polygons, and Distributions

Question 1

Histogram

Answer 1

A type of graphic illustration (of FIGURE) of a Frequency Distribution.

• Designed to present the data visually and thus easier to interpret and understand.
• In the Histogram below, frequencies are shown on the vertical axis (the ORDINATE).
• The scores (or score intervals) are shown on the horizontal axis (the ABSCISSA).
• You could also use percentages by placing percentages instead of frequencies on the vertical axis.

GUIDELINES for preparing histograms:

1. Draw the histogram on graph paper.
2. Place the scores on the horizontal axis, increasing as you move to the right.
3. Label the histogram with a number and a brief title (i.e., caption). Note that because the histogram is a statistical figure (i.e., a drawing or graph), it is called a “figure” and not a “table”.
4. If there is a score interval with a frequency of zero, a space MUST appear there. Notice the space for the interval 22-26 in the histogram, which has a frequency of zero.

Question 2

Frequency Polygon

Answer 2

A Frequency Polygon is another way to graphically illustrate Frequency Distributions. It is created the same way that a Histogram is created, except that instead of putting a column at each score (or score Interval), you put a mark where the top of the column would be indicating the FREQUENCY at that score and then connect all the marks when you are done plotting them. This produces an image that looks like a polygon.

GUIDELINES for Constructing Frequency Polygons:

1. List the frequencies (f) on the vertical axis (i.e., the ORDINATE or y-axis) and label the axis with an ‘f’
2. List the scores on the horizontal axis (i.e., the ABSCISSA or x-axis) and label the axis (in this case, “Depression Scores”). Begin with one score lower than any obtained by a subject and end with one score higher than any obtained. This will make the polygon rest, or anchor, on the horizontal axis. Dashed lines have been used to indicate that the anchor scores of 14 and 23 were not obtained by any of the examinees.
3. Put a dot on the graph paper at the point at which each score intersects each frequency. (For instance, in Figure 1, a dot was placed where a score of 15 intersects a frequency of 1 ). Then connect the dots with a ruler.
4. Be sure to connect dots that are on the horizontal axis. For instance, in Figure 1, there is a frequency of 0 (zero) at a score of 16. At that point, the line drops to the horizontal axis.
5. Label the polygon with a number and a brief title (i.e., caption).

Question 3

Frequency Polygons with Score Intervals (groups) and Curves

Answer 3

1. • 5. See Frequency Polygons and then take this 6th step when using SCORE INTERVALS.
2. For SCORE INTERVALS (groups), you must first determine the MIDPOINT of each interval. For instance, 40 is the midpoint (i.e., middle score) of the 39-41 score interval. Use these MIDPOINTS to represent each SCORE INTERVAL.

* With large samples, the lines on a frequency distribution usually will be fairly smooth-unlike the below examples, in which the polygon is jagged. When a smooth polygon emerges, it is usually referred to as a CURVE. Types of curves are described in the next section.

Question 4

Normal Curve – Shapes of Distributions

Answer 4

After a set of scores has been organized from lowest to highest, it is referred to as a DISTRIBUTION.

* One of the best ways to see a distribution’s shape is to create a frequency polygon. When the number of cases is large, the polygon usually has a smooth shape, often referred to as a CURVE.

* The most important shape is that of the NORMAL CURVE-often called the BELL-SHAPED CURVE.

A NORMAL DISTRIBUTION is SYMMETRICAL around its MEAN (or average) of all the scores in the distribution.

The NORMAL CURVE is IMPORTANT for two reasons:

1. It is a shape often FOUND IN NATURE. For instance, the heights of women in a large population are NORMALLY DISTRIBUTED. There are small numbers of very short women (which is why the curve is low on the left), there are many women of about average height (which is why the curve is high in the middle), and there are small numbers of very tall women (which is why the curve is low on the right).

* Another example: The distribution of average annual rainfall in Los Angeles over the past 110 years is approximately normal. There are a very small number of years in which there was extremely little rainfall, many years with about average rainfall, and a very small number of years with a great deal of rainfall.

2. The second reason the normal curve is important is that it is used in inferential statistics, a topic that is covered in Part B of this book.

Question 5

Skewed Distributions – Shape of Curves

Answer 5

While Normal Distributions are SYMMETRICAL around the MEAN, some distributions are SKEWED to one side or the other

A SKEWED DISTRIBUTION is UNBALANCED because of either some very high or very low scores. For instance, when the distribution of income for a large population is plotted on a polygon, it has a POSITIVE SKEW (The top CURVE below)

* Even though the curve appears to lean to the LEFT, it is actually the LARGE TAIL to the RIGHT that marks the curve as being POSITIVELY SKEWED. So remember to look at the tail in order to determine the skew: Larger tail to the right = Skewed to the right = POSITIVELY Skewed.

* This Positive skew in incomes indicates that there are large numbers of individuals with relatively low incomes. Thus, the curve is high on the left (where the lower incomes lie). The curve drops off dramatically to the right, forming a tail on the right. This tail is created by a small number of individuals with very high incomes.

* Remember: Skewed distributions are named for their TAILS. On a number line, POSITIVE numbers are to the RIGHT, hence the term POSITIVE SKEW.

NEGATIVE SKEW (The bottom CURVE below)

For everything you know about a positive skew, the opposite is true for a Negative Skew.

A NEGATIVE SKEW would be formed, for instance, if you tested a very large population of recent nursing school graduates on basic nursing skills, a distribution with a negative skew should emerge. There should be a large number of nurses with high scores, but there should also be a tail to the left created by a small number of nurses who, for one reason or another (such as being ill the day the test was administered), did not perform well on the test.

* Remember: Skewed distributions are named for their TAILS. On a number line, NEGATIVE numbers are to the LEFT, hence the term NEGATIVE SKEW.

Question 6

Bimodal Distribution – Shapes of Distributions

Answer 6

BIMODAL DISTRIBUTIONS have two high points even if the two high points are not equal in height.

* Such a curve is most likely to emerge when HUMAN INTERVENTION or some RARE event has changed the composition of a population. An example of human intervention is the decision of a school board to establish a school to which only high achievers and low achievers are admitted because the school is to be a model of peer tutoring in which the high achievers tutor the low achievers. The distribution of scores on an achievement test for students admitted to the school should be bimodal because the ‘average students that would normally form the bulk of the middle of a normal distribution are absent. Another example: When a war costs the lives of many young adults, the distribution of age after the war might be bimodal, with a dip in the middle.

* Bimodal distributions are much less common than normal distributions or skewed distributions.

* NOTE: The shape of a distribution should always be examined before proceeding with additional statistical analyses. The shape has important implications for determining WHICH AVERAGE to compute and which other STATISTICS are APPROPRIATE to compute.