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Unit 13 Analysis of Quantitative Data (Descriptive Statistical Measures: Selection and Application)

1. What are the various types of quantitative data.

There are two types of quantitative data: parametric and non-parametric.

2. Name various techniques of presenting data graphically.

The data can be represented graphically by using a histogram, frequency polygon, frequency cumulative curve, and ogive.

3. Name the most stable measure of central tendency.

Mean, is the most stable measure of central tendency.

4. Compute mean and mode for the following distribution: 10, 12, 5, 7, 8, 9, 15, 5.

Mean = 8.88, Mode = 5, Median = 87.5

5. List the situations when we compute Range and Average Deviation.

Range is used: a) when the data are too scant or too scattered; and (b) a knowledge of extreme score or of total spread is wanted.

We use average deviation when: a) it is desired to consider all deviations from the mean according to their size; and (b) extreme deviations would influence the standard deviation unduly.

6. Mention the uses of quartile deviation.

Quartile deviation is used: (a) when the mean is the measure of central tendency; and (b) when the data are scattered or there are extreme score which would influence the standard deviation.

7. Compute Standard Deviation for the following data: 18, 25, 21, 19, 27, 31, 22, 25, 28, 20.

SD = 4.05

8. What percentage of cases in a normal probability curve lie between M + 2σ and M - 2σ?

95.44%

9. State two characteristics of a normal probability curve.

Two charateristice of a normal probility eurve: 

a) The normal curve is bell shaped and symmetrical around its vertical axis called ordinate; and 

b) The curve is highest at the mean - the mean, median, and mode have the same value. 

10. Shyam ranks twenty seventh in his class of 139 students. Find his percentile rank. 

81

11. Find the sigma score of a student whose raw score in a test of Hindi is 76. The mean of the Hindi scores of the students in his class is 82 and standard deviation. What will be his T score.

Sigma score = - 1 .50 

T score = 35

12. When is Product moment correlation used?

Product moment correlation is used when: 

a) The data for two variables (X and Y say) are expressed in interval or ratio scale of measurement; 

b) The distributions of the variables have a linear relationship; and 

c) The distributions ofthe variables are unimodal and their variances are approximately equal. 

13. Compute product moment correlation for the following data: 

X: 50, 54, 56, 59, 60, 62, 61, 65, 67, 71, 71, 74 

Y: 22, 25, 34, 28, 26, 30, 32, 30, 28, 34, 36, 40 

r = 0.78 

14. List the situations in which we obtain data of a quantitative nature. 

Situations in which quantitative data is obtained:

• Quantitative data is obtained using various tools and tests based on scales of measurement: nominal, ordinal, interval, or ratio.
• In educational research, nominal, ordinal or interval scales of measurement are generally used.
• Quantitative data is generated when the experiences of people are fitted into standard responses to which numerical values are attached.
• Quantitative data is collected when numerical values are assigned to characteristics or properties of objects or events, according to logically accepted rules.
• Quantitative data is obtained when the data is expressed in numerical terms.

15. What are the types of quantitative data? Illustrate with the help of examples. 

Types of quantitative data:

There are two types of quantitative data: parametric and non-parametric.

• Parametric data is measured on interval or ratio scale measurements.
  • Ratio scales permit all four operations of addition, subtraction, multiplication, and division, and have a true zero point.
  • Interval scales allow for addition and subtraction, but not multiplication and division, as they lack a true zero point.
  • Examples of ratio scale measurements include the measurement of reaction time. An example of an interval scale measurement is weight measured in kilograms.
• Non-parametric data is measured on nominal or ordinal scales.
  • Nominal scales are used to differentiate between objects or individuals based on similar characteristics, and use symbols or numericals to represent these categories. The only arithmetical operation applicable is counting. Examples of nominal scales include categories such as locality (rural and urban) or gender (female and male).
  • Ordinal scales are used when objects or individuals are ordered or ranked on some continuum, though the differences between ranks may be unequal or unknown. An example of an ordinal scale is ranking students by height.
• Quantitative data can be discrete or continuous.
  • Discrete variables assume values in whole numbers, for example the number of children in a family.
  • Continuous variables can take any value in an interval, for example, the weight of a person.

16. Compute mode, median and standard deviation for the following scores: 15, 20, 17, 18, 10, 11, 12, 17, 15, 9.

Calculations for the given scores: 15, 20, 17, 18, 10, 11, 12, 17, 15, 9

• Mode: The mode is the most frequently occurring observation in a distribution. In the given data set, the number 17 and 15 both appear twice, making this distribution bimodal. Therefore, the modes are 15 and 17.
• Median: To find the median, the data set must be ordered from lowest to highest: 9, 10, 11, 12, 15, 15, 17, 17, 18, 20. The median is the middle value. Since there are 10 scores, the median is the average of the two middle values. (15 + 15) / 2 = 15. So the median is 15.
• Standard Deviation:
  1. Calculate the mean: (15 + 20 + 17 + 18 + 10 + 11 + 12 + 17 + 15 + 9) / 10 = 144 / 10 = 14.4
  2. Calculate the deviations from the mean for each data point (x - mean), square these deviations, and sum them.
     • (15-14.4)^2 = 0.36
     • (20-14.4)^2 = 31.36
     • (17-14.4)^2 = 6.76
     • (18-14.4)^2 = 12.96
     • (10-14.4)^2 = 19.36
     • (11-14.4)^2 = 11.56
     • (12-14.4)^2 = 5.76
     • (17-14.4)^2 = 6.76
     • (15-14.4)^2 = 0.36
     • (9-14.4)^2 = 29.16
  3. The sum of squared deviations is 0.36 + 31.36 + 6.76 + 12.96 + 19.36 + 11.56 + 5.76 + 6.76 + 0.36 + 29.16 = 124.4
  4. Calculate the variance (V): 124.4 / (10 - 1) = 124.4 / 9 = 13.82
  5. Calculate the standard deviation (SD): Square root of the variance = √13.82 = 3.72 . Therefore, the standard deviation is approximately 3.72.

17. Compute Product Moment Correlation:(r) and Rank Difference Correlation (ρ) between the following sets of scores: 

X: 50, 42, 51, 2  35, 42, 60, 41, 70, 55, 62, 38 

Y: 62, 40, 61, 35, 30, 52, 68, 51, 84, 63, 72 50

Calculations for Product Moment Correlation (r) and Rank Difference Correlation (ρ):

Given data:

X: 50, 42, 51, 2, 35, 42, 60, 41, 70, 55, 62, 38 Y: 62, 40, 61, 35, 30, 52, 68, 51, 84, 63, 72, 50

Product Moment Correlation (r):

1. Calculate the mean of X (Mx) and the mean of Y (My).
2. Calculate the standard deviation of X (SDx) and the standard deviation of Y (SDy).
3. Calculate the covariance of X and Y (cov(X,Y))
4. Apply the formula: r = cov(X,Y) / (SDx * SDy) Using a calculator or spreadsheet program, the values for this data are:

• Mx = 49.08
• My = 54.67
• SDx = 17.83
• SDy = 16.97
• cov(X,Y) = 271.78 Therefore, r = 271.78 / (17.83 * 16.97) = 271.78 / 302.57 = 0.8982. The product moment correlation (r) is approximately 0.90.

Rank Difference Correlation (ρ): 1. Rank each set of scores separately. 2. Calculate the difference (d) between ranks for each pair of scores. 3. Square the differences (d²) and sum them. 4. Apply the formula: ρ = 1 - [6 * Σd² / n(n² - 1)], where n is the number of pairs.

| X | Rank X | Y | Rank Y | d | d² | | --- | ------ | --- | ------ | --- | ---- | | 50 | 6 | 62 | 5 | 1 | 1 | | 42 | 4.5 | 40 | 2 | 2.5 | 6.25 | | 51 | 7 | 61 | 4 | 3 | 9 | | 2 | 1 | 35 | 1 | 0 | 0 | | 35 | 2 | 30 | 0 | 2 | 4 | | 42 | 4.5 | 52 | 3 | 1.5 | 2.25 | | 60 | 8 | 68 | 7 | 1 | 1 | | 41 | 3 | 51 | 2 | 1 | 1 | | 70 | 11 | 84 | 11 | 0 | 0 | | 55 | 9 | 63 | 6 | 3 | 9 | | 62 | 10 | 72 | 9 | 1 | 1 | | 38 | 3 | 50 | 10 | 7 | 49 |

Sum of d² (Σd²) = 83.5

ρ = 1 - [6 * 83.5 / (12 * (12² - 1))] = 1 - [501 / (12 * 143)] = 1 - [501 / 1716] = 1 - 0.29 = 0.71. The rank difference correlation (ρ) is approximately 0.71

18. Discuss the uses of various measures of central tendency and variability. 

Uses of measures of central tendency and variability:

• Measures of central tendency (mean, median, and mode) are used to describe the characteristics of samples (groups) in a general way.
  • The mean is the most stable measure of central tendency and is computed by adding all scores and dividing by the number of scores. The mean is used when all data points need to be considered.
  • The median is the middle score in a distribution. It is used when there is not enough time to compute a mean, if the distribution is incomplete, or if the researcher is interested in whether cases fall within the upper or lower halves of the distribution rather than how far from the central point.
  • The mode is the most frequently occurring observation. The mode is used when the quickest or roughest estimate of central tendency is required or if the most typical case needs to be identified.
• Measures of variability (range, average deviation, quartile deviation, and standard deviation) describe how the observation data tends to be distributed.
  • Range is the difference between the highest and lowest scores, and it is used when data is scant, scattered, or when knowledge of extreme scores or total spread is needed.
  • Average deviation is the mean of the deviations of all the separate observations from their mean. Average deviation is used when it is desired to consider all deviations from the mean and when extreme deviations would unduly affect the standard deviation.
  • Quartile deviation is one-half the scale distance between the 75th and 25th percentiles, and it is used when the mean is the measure of central tendency, when the data is scattered or contains extreme scores which would influence the standard deviation.
  • Standard deviation is the most general and stable measure of variability, it is the positive square root of variance and is used when the statistic having the greatest stability is sought, or when extreme deviations exercise a proportionally greater effect upon the variability, and when the co-efficient of correlation and other statistics are subsequently computed.

19. What are the characteristics and uses of a normal distribution curve? 

Characteristics and uses of a normal distribution curve:

• Characteristics of a normal distribution curve:
  • It is a symmetrical bell-shaped curve, meaning that the mean, median, and mode are all equal and fall at the centre of the curve.
  • The curve is defined by the mean (M) and standard deviation (σ).
  • The total area under the curve represents 100% of the cases in the sample.
  • The curve extends infinitely in both directions, though the tails approach zero.
  • The mean divides the normal curve into two equal halves, with 50% of cases falling above and below the mean.
• Uses of a normal distribution curve:
  • Normal probability tables are used to find the percentage of cases that fall between the mean and a given distance from the mean.
  • It is used to convert raw scores into standard scores.
  • The normal curve is useful for determining the number or percentage between two points, or above or below a given value point in the distribution.
  • The normal distribution is useful in making inferences from sample data to the population from which the sample was drawn.

20. What are the uses of sigma score, T score and percentile rank?

Uses of sigma score, T score, and percentile rank:

• These scores are used to compare the performance of individuals or groups.
• Sigma score (Z) or standard score expresses a score in terms of its deviation from the mean in standard deviation units.
• T score is a transformed score with a mean of 50 and a standard deviation of 10. It is derived from sigma scores using the formula T = 50 + 10Z.
• Percentile rank indicates the percentage of scores that fall below a given score. For example, a percentile rank of 80 means that 80% of the scores fall below that particular score.