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2.2 Frequency Distributions
A frequency distribution is an organized tabulation of the number of individuals or frequency of occurrence in each category on a scale of measurement.
Frequency Distribution is a table constructed to show how many times a given score or group of scores occurred in a set of data. A simple frequency (ungrouped frequency) distribution (see Table 2.2.1) is the ordering of the scores of a distribution from highest to lowest scores in a table with their corresponding frequencies. When scores are grouped into intervals showing how many scores occurred in each interval, this is called a grouped frequency distribution (Table 2.2.2). Apparent Limits are the limits displayed in a grouped frequency table (Table 2.2.2, col. 1); these limits give a reasonable range between which groups of data exist. Real Limits (for continuous data – measurements or observations that depend upon the accuracy of the measuring instrument) of any interval extend from ½ unit below and above the apparent lower and upper limits respectively. The real limits of the “20 – 30” interval are 19.5 and 30.5. The real lower limit is designated L and the real upper limit is designated U. The Midpoint , MP , of an interval is its exact center. The MP of any interval is found by adding the apparent upper limit to the apparent lower limit and dividing by 2. The MP of the “20 – 30” interval is 25. Interval size is denoted by the symbol i ; it is the distance between the real lower limit and the real upper limit. The interval size is determined by subtracting L from U. It is recommended that you group data so that there are between 8 and 15 intervals.
Frequency ( f ) simply indicates how many scores are located or counted in each interval. Number , N , is the number of scores in a distribution or the total of all the frequencies for all the intervals. N is also called the sample size. The following are some outputs of frequency distributions from various statistical programs and Excel spreadsheets.
Table 2.2.1 Simple Frequency Distribution of 9th Graders (ODE)
9 th^ Graders (^) Frequency ( f ) Percent 10098 11 1.061. 95 1 1. 8685 11 1.061. 84 2 2. 8382 21 2.131. 8180 21 2.131. 79 1 1. 7877 32 3.192. 75 4 4. 7473 35 3.195. 72 3 3. 7169 34 3.194. 68 5 5. 6766 33 3.193. 65 3 3. 6463 31 3.191. 62 2 2. 6160 42 4.262.
Figure 2.2. 1 SPSS frequency procedure: Analyze -> Charts -> Select Histograms Tony and Frequency Distributions Tony: “Given a set of data, how best to construct a grouped frequency distribution? Rose: “The concept of frequency distribution is not complex; however, the mechanics of construction a grouped and sometimes a simple frequency distribution table
of graph is often complex for most students.” “So it is best illustrated with the statistics tutorials and the Excel spreadsheet program.” The mode is readily computed from a frequency distribution; the most frequent score or the midpoint of group interval.
Table 2.2. Grouped Frequency SPSS Output Scores (^) Frequency 20 – 29 (^1) 30 – 39 (^3) 40 – 49 (^5) 50 – 59 (^18) 60 – 69 (^30) 70 – 79 (^24) 80 – 89 (^10) 90 – 100 (^2) 100 – 109 (^1)
Figure 2.2.2 Histogram of grouped frequency.
Table 2.2.4 Output from Stat4U after Recoding FREQUENCY ANALYSIS BY BILL MILLER (Stats4U) Frequency Analysis for passFreqMP UP TO FREQ. PCNT CUM.FREQ. CUM.PCNT. %ILE RANK
24.5034.50 34.5044.50 13 0.010.03 1.004.00 0.000.04 0.010. 44.5054.50 54.5064.50 185 0.050.19 (^) 27.009.00 0.100.29 0.070. 64.5074.50 74.5084.50 3024 0.320.26 57.0081.00 0.610.86 0.450. 84.5094.50 (^) 104.5094.50 102 0.110.02 91.0093.00 0.970.99 0.910. 104.50 114.50 1 0.01 94.00 1.00 0.
(^0) 0.00 20.00 40.00 grade9th 60.00 80.00 100.
5
10
15
20
25
30
Frequency
Mean = 61.2766 Std. Dev. = 13.5388 N = 94
Histogram
Cumulative Distributions Cumulative distributions, like proportional and percentage distributions, form the basis for comparative interpretation of data. Cumulative frequency distribution represents the number of cases up to and including a specific score or value. There are three forms to the cumulative distributions (listed, ungrouped, grouped). Listed and Ungrouped Cumulative Distributions The dataset of listing of data is such that no range or values of scores occurs more than once. Table 2.2.4 shows the cumulative frequency distribution table. The cumulative frequency, Cf column shows that for score value of 30 or less, there is only one case. For score value 34, there are 5 cases. All cumulative frequency tables are form by cumulating the frequency for each score from bottom to top (remember that in a frequency table the raw scores or groups intervals are listed highest at top of table and lowest at bottom). Table 2.2.5 shows the ungrouped cumulative frequency distribution, we follow the same rule as we did for determining the cumulative listed frequency. Observed that the highest cumulative frequency is equal that of the total frequency, n.
Table 2.1. 4 Listed Cumulative Distribution
Xi f i Cf
n = 6
Table 2.1.5 Ungrouped Cumulative Distribution
Xi f i Cf
n = 21
It is a simple step to go from a cumulative frequency distribution to a cumulative percentage or proportional distribution. To construct a cumulative percentage or proportional distribution, it is necessary to first construct a cumulative frequency distribution. Below is a guide to construct a cumulative percentage or proportional distribution for a listed frequency or ungrouped frequency distribution. Construction Guide : Proportional and Percentage Cumulative Distribution (ungrouped) The cumulative proportion for each score Cp : C (^) p Cn^ f , where Cf is frequency for a given score and n is total frequency
The cumulative percentage for each score, CP : CP Cn^ f 100
Xi f i Cf Cp CP(%)
n = 21 Max( Cp ) = 1.0^ Max( CP ) = 100%
We could make the following statement from the cumulative distribution table, 61.9% of scores are less than or equal to (=<) X = 33.
Construction of cumulative percentage or proportional distribution for grouped frequency tabulation is similar to constructing one from an ungrouped frequency distribution. When data are grouped into scale intervals or classes, we use the midpoint of the class or interval when we reference the frequency, percentage, or a proportion. For
Although the cumulative percentage distribution gives the percentage of all scores that fall at or below an observed value or score, it is often necessary to compute the score for a specific cumulative percentage. For example, in the previous example, we found that 71.4% of scores were at or below the value of 49. We may need to find the score at which 70%, or 90%, or 50% (median), etc., below which all scores falls. To find the scores at which a given percentage of the distribution fall below for percentage points not readily obvious from the cumulative percentage distribution, we need to do additional computation; we need to find the Percentile.
Percentile from Cumulative Percentage Distribution
Percentile is the score at which a certain percent (%) of the scores is less than or equal to.
The percentile of a cumulative percentage distribution or a transformed frequency distribution is determined by the following formula:
P % LL (^) i n^ p^ f (^) i C^ f I
where P% = any specified percentile point LL (^) i = the exact lower limit or real lower limit of the group interval containing Percentile point, P% n (^) p = number of scores or cases comprising the specified percentage for n. n (^) p p n , where p is decimal equivalent to % and n is total frequency Cf = cumulative frequency up to but not including the percentile interval f (^) i = frequency within the percentile interval I = group interval size or class size
Construction Guide : Computing Percentile Points Percentile from cumulative distribution P % LL (^) i n^ p^ f (^) i C^ f I 43.5 10.5 3 10 3 43. P% = any specified percentile point LL (^) i = the exact lower limit or real lower limit of the group interval containing Percentile point, P% n (^) p = number of scores or cases comprising the specified percentage for n. n (^) p p n , where p is decimal equivalent to % and n is total frequency Cf = cumulative frequency up to but not including the percentile interval I^ f i = group interval size or class size^ = frequency within the percentile interval Xi f (^) i Cf Cp CP(%) 50-52 6 21 1.00^ 100. 47-49 2 15 0.714^ 71. 44-46 3^13 0.619^ 61. 41-4338-40 14 109 0.4760.429^ 47.642. 35-37 (^) n = 21 5 5 0.238^ 23.
Question: Find the score at which 50% of all scores fall (the median) Step 1: Compute number of cases equal to 50% of n, np = (0.50)(21) = 10. Step 2: Using Cf , find the interval at which 50% falls, percentile interval or median interval here: 44 – 46 Step 3: Find LL (^) i , the exact lower limit of the percentile interval, LL (^) i = 43. Step 4: Find the number of cases up to but not including those in the percentile interval Cf = 10 Step 5: Subtract ( n (^) p - Cf ) = 10.5 – 10 = 0. Step 6: Divide step 5 by number of case within percentile interval, f (^) i : 0.5/3 = 0. Step 7: Multiply Step 6 by interval size, I = 3 , 0.1667/3 = 0. Step 8: Add Step 7 and Step 3: 43.5 + 0.0556 = 43.556 or P 50 = 43.
Construction Guide : Computing Mean from Grouped Data Mean from frequency:
mean = X fn^ i^ X i^92115 43.
Xi Midpoint (New X) f f X 50-52 (^51 6 ) 47-49 (^48 2 ) 44-46 (^45 3 ) 41-43 (^42 1 ) 38-40 (^39 4 ) 35-37 (^36 5 ) n = 21 (^) f (^) i X (^) i 915
Step 1: List the midpoint for each class interval, new X Step 2: Multiply ( f )(Midpoint) or f X Step 3: Add f X column, f X = 915 Step 4: Add f column f = 21 Step 4: Divide Step 3 by Step 4, 915/21 = 43.
