standard deviation: an introduction
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The document discusses standard deviation (SD) as a measure of variability in a data set, detailing its definition, significance in statistics, and characteristics such as its relationship with mean values. It provides methods for computing SD for ungrouped and grouped data along with examples. Additionally, it addresses how SD indicates data dispersion and its application in statistical techniques.
standard deviation: an introduction
- 1. Standard Deviation Dr Rajesh Verma Asst. Prof (Psychology) Govt. College Adampur, Hisar, HaryanaPC Mahalanobis, Indian Legend
- 2. Meaning-Cum-Definition It’s a quantity that indicates by how much the members of a group differ from the mean value for the group. Standard deviation is and index of degree of dispersion and an estimate of the variability in the population from which the sample is drawn (Guilford & Fruchter, 1976).
- 3. Introduction SD is also known as Coefficient of Variation (CV). It’s a kind of special average of all the deviation from the group mean. Special average means that SD is beyond the simple arithmetic mean. Along with the total range, semi- interquartile range, average deviation it is the most commonly used single number to indicate variability among a data set. Variability is the scatter or spread of the separate scores around their central tendency (Garret, 2014). SD is the most stable index of variability which is computed using squared deviations that are taken from mean only i.e. neither median nor mode.
- 4. Characteristics (i) High value of SD means that most of the scores are away from group mean and vice versa. (ii) It is the index of spread of scores about the mean that involves every observation. (iii) Greater SD than mean indicates that the distribution have majority of low values. (iv) Lower SD than mean indicates that the distribution have majority of high values. (v) It can also indicate that whether the distribution has dispersion of extreme values or has central tendency.
- 5. (vi) It is expressed in the same units as the data. (vii) It is possible to calculate the combined standard deviation of two or more groups. (viii) The SD is considered most appropriate statistical tool for comparing the variability of two or more groups. (ix) It is used as unit of measurement in normal probability distribution. (x) SD forms the base for several statistical techniques such skewness, kurtosis, ANOVA etc.
- 6. How much SD is Good or Bad No amount of SD is good or bad because it’s just an indicator of spread of scores. If someone is interested to see the closeness of scores around the mean than SD ≤ 1 is good, on the other hand if someone is interested in larger spread than SD ≥ 1 is good. Across books and manuals it has been accepted that SD ≥ 1 is the indicator of high variation among the scores and SD ≤ 1 indicates low variation. It’s a kind of rule of thumb. In fact it depend upon the data set and objectives of the researcher.
- 7. Computation of SD For Ungrouped Data Let us take an hypothetical data. Ex – 12, 54, 32, 51, 24, 58, 21, 43, 31, 48 Step I – Calculate the mean. Mean ( 𝒙) = ∑𝒙 𝑵 Where, x = Score N = total no. of score Therefore, 12+54+32+51+24 +58+21+43+31+48 = 374 𝒙 = 𝟑𝟕𝟒 𝟏𝟏 = 34
- 8. Step II – Calculate deviation (d) of each score from the mean. (i) It is achieved by subtracting group mean from each score, (ii) Square the each Deviation, (iii) Sum up squared Deviations, Step III – Calculate SD by using the following formula σ = ∑𝒅 𝟐 𝑵 => 𝟐𝟑𝟐𝟕 𝟏𝟏 => 𝟐𝟏𝟏. 𝟓𝟒𝟓 = 14.545
- 9. SD of Grouped Data Let us take Hypothetical Data as shown in adjacent table Step I - Calculate the mean For calculation of mean we must follow the following few steps
- 10. Steps of calculating mean (i) Find the midpoints of CI (x) (ii) Multiply midpoints (x) with their respective frequency (f) (iii) Add fx (iv) Divide it by the sum of frequencies i.e. 50 mean = ∑𝒇 𝒙 𝑵 = 𝟏𝟔𝟓𝟎 𝟓𝟎 = 33
- 11. Step II-Subtract mean from the mid points i.e. 𝒙 − 𝒙 Step III-Square these values which are denoted by (x`) Step IV-Multiply x` with their respective frequencies (fx`) Step V-Add fx` Finally substitute the values in the formula σ = ∑𝒇𝒙` 𝑵 = 𝟓𝟓𝟏𝟐 𝟓𝟎 = 𝟏𝟏𝟎. 𝟐𝟒 (𝒔 𝟐 = 110.24) = 10.499
- 12. References: 1. https://dictionary.apa.org/quartile-deviation. 2. Guilford, J. P. and Fruchter, B. (1978). Fundamental Statistics in Psychology and Education, 6th ed. Tokyo: McGraw-Hill. 3. https://todayinsci.com/M/Mahalanobis_Prasanta/ MahalanobisPrasanta-Quotations.htm. 4. Garrett, H. E. (2014). Statistics in Psychology and Education. New Delhi: Pragon International. 5. Levin, J. & Fox, J. A. (2006). Elementary Statistics. New Delhi: Pearson.