Chapter 2

Prepared ByMath Coordinator

Salwa KamelMath Qualification from Cambridge University

Data Description

Data Description

Measures of Central Tendency for:

•Ungrouped data

• Ungrouped frequency distribution

•Grouped data

Objectives:

By the end of this chapter the students will be able to:

1) identify and calculate three measures of central tendency: mean, median

and mode.

2) Explore how measures of central tendency are affected by changes in the

data.

3) Explain the concept of central tendency.

4) Identify and compute the arithmetic mean.

5) Compute and interpret the weighted mean.

6) Determine the median.

7) Identify the mode.

4

SOME IMPORTANT DEFINITIONS

5

POPULATION AND SAMPLE

POPULATION: A population consists of an entire set of objects, observations, or scores that have something in common. For example, a population might be defined as all males between the ages of 15 and 18.

SAMPLE: A sample is a subset of a Population Since it is usually impractical to test every member of a population, a sample from the population is typically the best approach available.

6

PARAMETER AND STATISTIC

PARAMETER: A parameter is a numerical quantity measuring some aspect of a population of scores. For example, the mean is a measure of central tendency in a population.

STATISTIC: A "statistic" is defined as a numerical quantity (such as the mean calculated in a sample).

Data Summarization

To summarize data, we need to use one or two parameters that can describe the data:

Measures of Central tendency which

describes the center of the data

and the Measures of Dispersion, which show how the data are scattered around its center.

Measures of Central Tendency

Definition: A Measure of Central Tendency has been defined as a statistic calculated from a set of observations or scores and designed to typify or represent that series.

It is also defined as the tendency of the same observations or cases to cluster about a point, with either to an absolute value or to a frequency of occurrence; usually but not necessarily, about midway between the extreme high and the extreme low values in the distribution.

Measures of Central Tendency

In daily life we are using the words Average, Middle and Most

Frequent quite often, which can be statistically described as

follows:

• Mean

• Median

• Mode

The Mean

Definition: The arithmetic mean or simply the mean is the average of a group of measures.

Characteristics of the mean

1. The arithmetic mean, or simply mean is the center of gravity

or balance point of a group of measures.

2. The mean is easily affected by a change in the magnitude of any of the measures.

Characteristics of the Mean

3. The mean is the most reliable measure of central tendency because it is always the center of gravity of any group of measures.

Uses of the Mean

Compute the mean when

1. the mean of a group of measures is needed.2. the center of gravity or balanced point of a group of

measures is wanted.3. every measure should have an effect upon the measure of

central tendency.

Definition: The arithmetic mean or simply the mean of a data set is the sum of the values divided by the number of values. That is, if X1, X2, . . . , XN are the individual scores in a population of size N, then the population mean is defined as:

Definition: If X1, X2, . . . , Xn are the individual scores in a sample size n, then the sample mean is defined as:

N

X

X

n

X

X

Population Mean

For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values:

EXAMPLE – Population Mean

There are 42 exits on I-75 through the state of Kentucky. Listed below are the distances between exits (in miles).

Why is this information a population? What is the mean number of miles between exits?

EXAMPLE – Population MeanThere are 42 exits on I-75 through the state of Kentucky. Listed below are the distances between exits (in miles).

Why is this information a population?

This is a population because we are considering all the exits in Kentucky.

What is the mean number of miles between exits?

Sample Mean

• For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values:

EXAMPLE – Sample Mean

Properties of the Mean

• subgroup means can be combined to come up with a group mean

• easily affected by extreme values

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5Mean = 6

Example:

• The hemoglobin levels in the blood of eight female

volunteers are 13.3, 11.1, 17.4, 11.7, 12.7, 15.2, 12.3 and 11.9 g/dl .– What is the mean hemoglobin level?

The Weighted Mean

Definition: The Weighted Mean is a variation of the arithmetic mean which assigns weight to the individual scores in a data set.

where - the weighted mean

- the weight

- the individual scores

- number of cases

W

XW

XW

XWW

X

n

EXAMPLE – Weighted Mean

The Carter Construction Company pays its hourly employees $16.50, $19.00, or $25.00 per hour. There are 26 hourly employees, 14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2 at the $25.00 rate.

What is the mean hourly rate paid the 26 employees?

The Median

Definition: The median is the middle most value in an ordered sequence of data.

Remark: The median is unaffected by any extreme observations in a set of data and hence, whenever an extreme observation is present, it is appropriate to use the median rather than the mean to describe a set of data.

Statistical Treatment: For an even number of observations:

22

2

2

nn XX

Md

Determine the median.

For an odd number of observations:

Example: A manufacturer of flashlight batteries took a sample of 13 from a day’s production and burned them continuously until they failed. The number of hours they burned were

342 426 317 545 264 451 1049

631 512 266 492 562 298.

Determine the median.

2

1 nXMd

Determine the median.

The Median

PROPERTIES OF THE MEDIAN1. There is a unique median for each data set.2. It is not affected by extremely large or small values and is therefore a

valuable measure of central tendency when such values occur.3. It can be computed for ratio-level, interval-level, and ordinal-level data.4. It can be computed for an open-ended frequency distribution if the

median does not lie in an open-ended class.

MEDIAN The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest.

Determine the median.

The Median

Properties of a Median

• may not be an actual observation in the data set

• can be applied in at least ordinal level• a positional measure; not affected by

extreme values

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5

EXAMPLES - Median

The ages for a sample of five college students are:21, 25, 19, 20, 22

Arranging the data in ascending order gives:

19, 20, 21, 22, 25.

Thus the median is 21.

The heights of four basketball players, in inches, are:

76, 73, 80, 75

Arranging the data in ascending order gives:

73, 75, 76, 80.

Thus the median is 75.5

The Mode

Definition: The mode is the value in a set of data that appears most frequently. It may be obtained from an ordered array.

Remark: Unlike the arithmetic mean, the mode is not affected by the occurrence of any extreme values. However, the mode is used only for descriptive purposes because it is more variable from sample to sample than other measures of central tendency.

Example: Consider the out – of – state tuition rates for the six – school sample from Pennsylvania.

4.9 6.3 7.7 8.9 7.7 10.3 11.7

The ModeMODE The value of the observation that appears most frequently.

Identify the mode.

Session 1.29

TEACHING BASIC STATISTICS

Mode

• occurs most frequently • nominal average• computation of the mode for ungrouped or raw

data

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode

Properties of a Mode

• can be used for qualitative as well as quantitative data

• may not be unique• not affected by extreme values• may not exist

Example:

Five cancer patients, the first survived for two months, the

second survived for six months, the third and fourth

patients survived for four months and the last one survived

for 30 months. Find the mean, median and the mode.

Solution:

Example

• A psychologist is working with a group of 10 children who are being treated for autism,(a mental disorder in which the individual avoids contact with other people and often cannot speak). The following data are obtained according to the age in months at which autistic behavior was first observed.

1 6 8 3 2 3 14 24 7 4

– Find the mean, median and mode.

Measures of central tendency for ungrouped frequency distribution

Sometime it is hard for us to tell the information from a table full of row data. Therefore the researchers organize the data by constructing a frequency distribution.

Calculation of Mean, Median, Mode For frequency Distribution Data

In case of frequency distribution data we calculate the mean by this equation:

x = ∑ fx nwhere f = frequency

Calculation of Mean, Median, Mode For frequency Distribution Data

for example : we want to calculate the mean incubation period of this group.

Example:

–Psychological studied of memory in human make use of random word lists. Each subject was given 5 minutes to study a list of 15 words and was then asked to list as many of these words as can be recalled. The following data were obtained.

14, 6, 12, 7, 5, 7, 11, 6, 5,1110, 6, 9, 9, 12, 5, 7, 14, 5, 6 5, 7, 7, 11, 5, 9, 4, 12, 10,7Find the mean median and the mode.

X f4 15 66 47 69 3

10 211 312 314 2

The mean, median and mode for ungrouped frequency distribution

X f Cumulative frequency (F)

fx

4 1

5 6

6 4

7 6

9 3

10 2

11 3

12 3

14 2

Total

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he m

edian

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